lprint(`Reading menuF ..`);
Menu:=proc() local i,l,x,y;
#Print details of apecs command specified by args
  l:=proc();lprint(args);end:
  if nargs=0 then 
    l(`Menu, unlike menu, requires the name of an apecs command`);
    l(`e.g. Menu(eadd) or Menu(Eadd), even Menu(Menu)`);RETURN();
  fi;
  x:=args:i:='i':
  if member(x,[AgM,agM,'Allp','allp',App,app,Comb,comb,Eadd,eadd,'Eadp',\
'eadp',Ein0,ein0,ein,Ein,Ell,ell,Emod,emod,Emods,emods,Ford,ford,\
Gcub,gcub,Genj,genj,Ht,ht,Sub,sub,Isog,isog,'Menu','Mulp','mulp',Mult,mult,\
Om,om,Omp,omp,Quar,quar,Quar0,quar0,Rk,rk,Rk0,rk0,Seek,seek,Sfe,sfe,Sh\
a,sha,Trcw,trcw,Trwc,trwc,Tw,tw,FnL,fnL,Mest,mest,Trf,trf,On,on,'Onp','onp',\
Rk1,rk1,Neg,neg,Bas,bas,Ind,ind,Lin,lin,Gram,gram,Abel,abel,Roha,roha,Absc,\
absc,'Negp','negp',Search,search,Seek1,seek1,Raf,raf,We,we,Div,div,Tri,tri\
,Velu,velu,'Exam','exam',Hdif_Sik,hdif_Sik,Hdif_Sil,hdif_Sil,RkNC,rkNC,Expr,\
expr,Store,store,Trans,trans,Cubic_fit,cubic_fit,Third_pt,third_pt,Weier_fit,\
weier_fit,Tor,tor,'Qfin','qfin',Minf,minf,Fact,fact,Crem,crem,Om2,om2,Subg,subg\
,Torq,torq,Ub_tor,ub_tor,Go,go,Fundunit,fundunit,Cld,cld,Isom,isom,Seekb\
,seekb,qv,`q.v.`,Quarm,quarm,Ypecs,ypecs,Zpecs,zpecs,Afac,afac,Heeg,heeg],i)
  then
    if i<3 then
l(`agM(x1,y1)=(x+y)/2`);l(`AgM(x1,y1)=NULL`);
l(`   where x1,y1 are positive reals (either integers, rationals or floats)`);
l(`The arithmetic-geometric mean (denoted agM by Gauss) of x1 and x2 is the`);
l(`common limit of x=lim(xn) and y=lim(yn) where x.(n+1)=(xn+yn)/2,`);
l(`y.(n+1)=sqrt(xn*yn). These procedures calculate until x and y agree to`);
l(`"Digits" digits of accuracy.`);
l(`The Maple global variable "Digits" can be set at any time during an apecs`);
l(`session, e.g. Digits:=150;. The apecs default value is 20.`);
    elif i<5 then
l(`allp(p1)=Np`);
l(`Allp(p1)=NULL`);
l(`Np=order of group of non-singular points of E mod p1`);
l(`Other variables: if E mod p1 is singular then qs=node or cusp and`);
l(`singular point=[xs,ys].`);
l(`These procedures work for any weierstrass curve with rational coeff.'s`);
l(`whose denominators are prime to p1, even one input via the command Ell`);
l(`(Ell doesn't put E in min. Weier. form - sometimes desirable).`);
l(`Here's how we calculate square roots mod p: srqt(f)=f^w*t^i mod p`);
l(`where, if p=2^k*u+1, u odd, w=(u+1)/2, t=r^u, r any quadratic`);
l(`non-residue mod p, and we try in succession i=0,1,.. (some i<2^(k-1)`);
l(`will work). E.g. when p=-1 mod 4, sqrt=f^((p+1)/4); when p=5 mod 8,`);
l(`sqrt=f^((p+3)/8)*{1 or 2^((p-1)/4)}. For other p one has to find an r`);
l(`but on average it's worth it since there are quite a few sqrt's to find.`);
    elif i<7 then
l(`app(args)=NULL`);l(`App(args)=NULL`);
l(`    where args is either x or x,y or [x,y]`);
l(`If args is x then app calls the procedure absc(x) to calculate y.`);
l(`This procedure joins [x,y] to the list RR (if its not already there)`);
l(`with the following provisos:`);
l(`  If [x,y] is not a point on E the procedure is aborted.`);
l(`  If K_=1, so E is defined over Q (see Menu(ell) for the defn. of K_),`);
l(`then [x,y] must be defined over Q and either be independent mod torsion`);
l(`of the points already in RR, or improve RR by replacing one of the points\
.`);
l(`Note that when K_=1 the members of RR are arranged in order of increasing`);
l(`canonical height. Any improvement to RR will be carried over to known`);
l(`isogenous curves when feasible.`);
l(`  If K_=T, so E is defined over the function field K=Q(T) and at least one`);
l(`of the coefficients a1,..,a6 is not in Q (N.B: the transcendental must`);
l(`be T, not another name), or if K_=m_, so K=Q(sqrt(m_) (see Menu(Qfin)),`);
l(`app checks that [x,y] is defined over K, then`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`makes a modest check that the point is not already in the group generated`);
l(`by RR and torsion. This involves calculating the torsion subgroup, if`);
l(`that hasn't already been done. If lin (see Menu(Lin)) indicates that`);
l(`it is independent, then it is joined to RR.`);
l(`   For all other K_, [x,y] is checked not to be of finite order (well`);
l(`at least not of order <11) and not to be an "obvious" lin. comb.`);
l(`of points already in RR (i.e., not in the auxilliary set uR). If so,`);
l(`it is joined to RR and uR is enlarged accordingly.`);
    elif i<9 then
l(`comb(args)=suM`);l(`Comb(args)=NULL`);
l(`    Find a linear combination of points on E:`);
l(`    args is a sequence of pairs n1,z1,n2,z2,.. , each ni in Z (pos. or n\
eg.)`);
l(`and each zi is a point on E in the form x,y or [x,y] or [x,y,n].`);
l(`These procedures calculate suM:=[xc,yc] which is the linear combination`);
 l(`n1*z1+n2*z2+.. ,with the convention O=[NULL,NULL]=[], and they invoke`);
 l(`on and mult. If some on(zi)=false then xc='xc' is returned.`);
 l(`However this on test is disabled if hastype(zi,float) is true.`);
    elif i<11 then
l(`eadd(args)=[x3,y3]`);l(`Eadd(args)=NULL`);
l(`  where args=z1,z2 and for i=1,2, zi=xi,yi or [xi,yi] or, if zi has finite`);
l(`order n, [xi,yi,n]`);
l(`Calculate [x3,y3]=z1+z2 on E with the convention that O=[NULL,NULL]=[].`);
l(`This procedure works "in general"; some examples are`);
l(`Ein(F14);Eadd(-121/4,117/8,-91/3,(132+4*sqrt(-3))/9);===>`);
l(`[x3,y3]=[-25,12+28*(-3)**(1/2)]`);
l(`n:='n';Ell(0,0,0,n^2-1,0);Eadd(1,n,0,0);===>[n^2-1,n-n^3]`);
l(`For the next example recall that al3 is the 3-division poly. -- see N\
ota();`);
l(`and for absc, see Menu(absc);`);
l(`Ein(1);sols:=fsolve(al3);for i to 2 do P.i:=absc(sols[i]):od:===>`);
l(`cur=A11 with approximate points P1, P2 of order 3, one real, the other`);
l(`with a complex y-coord. Thus eadd(P1,P2) is another (approx.) pt.`);
l(`of order 3. eadd(P1,[0,0]) has order 15 since [0,0] has order 5.`);
    elif i<13 then
l(`eadp(args)=[x3,y3]`);l(`Eadp(args)=NULL`);
l(`  where args=z1,z2 and for i=1,2, zi=xi,yi or [xi,yi] and zi are points on`);
l(`E mod p (they don't have to be rational points in E(Q)). The value of p`);
l(`must be preset.`);
l(`Calculate [x3,y3]=z1+z2 on E mod p with the convention that O=[NULL,N\
ULL]=[].`);
l(`These procedures work for any weierstrass curve with rational coeff.'s`);
l(`whose denominators are prime to p, even one input via the command Ell`);
l(`(Ell doesn't put E in min. Weier. form - sometimes desirable).`);
    elif i<15 then
l(`ein0(args)=NULL`);l(`Ein0(args)=NULL`);
l(`   args=a0,a1,a2,a3,a4,a6`);
l(`Input the curve V^2+a1*U*V+a3*V=a0*U^3+a2*U^2+a4*U+a6, make the`);
l(`transformation U=X/a0, V=Y/a0 and invoke ein. Also append this curve to`);
l(`the list bec and the transformations between this curve and the min. w\
eier.`);
l(`as calculated by ein to the list bic.`);
l(`bec and bic are used by the commands Trcw and Trwc.`);
l(`    An EXAMPLE illustrating the convenience of this command based on the`);
l(`internet posting of Apr. 26/00 by Andrej Dujella: He considers curves`);
l(`(*)         y^2=(ax+1)(bx+1)(cx+1)`);
l(`where a,b,c are rational numbers such that ab+1, bc+1 and ca+1 are`);
l(`squares, for example when b=-1/a and c=a-1/a. Then (*) becomes`);
l(`(**)        y^2=(1/a-a)x^3 + (a^2+1/a^2-3)x^2 + 2(a-1/a)x + 1.`);
l(`To investigate the curves for various values of a one could define`);
    print(`Duj:=proc(a) Ein0(1/a-a,0,a^2+1/a^2-3,0,2*(a-1/a),1);...;end;`);
l(`Dujella discovered that a=145/408 gives a curve E with torsion/Q of`);
l(`order 16 and rank(E(Q))=3. This is the curve in the table displayed by`);
l(`the apecs command Exam(4).`);
    elif i<17 then
l(`ein(args)=curve is, cur`);l(`Ein(args)=NULL`);
l(`   where cur is the catalog name of the curve inputted by this ein`);
l(`   command (the name is assigned by ein if the curve is new) and`);
l(`   args is one of`);
l(`     (i) five rational weierstrass coefficients as a sequence a1,a2,a3,a4\
,a6`);
l(`         or as a list [a1,a2,a3,a4,a6]`);
l(`     (ii) an existing catalog name, e.g. A11`);
l(`     (iii) an existing catalog number, e.g. 3 (which specifies C11)`);
l(`This is the "Elliptic curve/Q input" command: it stores the data of the p\
resent`);
l(`curve (if there is one) in the catalog and the stack, (unless switched to`);
l(`"storage off" -- see below; "storage on" is the default; the command N\
ota();`); 
l(`lists the variables stored) and makes the curve specified by args the`);
l(`present curve.In cases (ii) and (iii) apecs is now pointing to that c\
atalog`);
l(`curve. In case  (i) it finds the min. Weier. form/Z by the Laska-Kraus a\
lgor.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`and, if the curve isn't itself min., checks whether the min. form is in t\
he`);
l(`catalog and updates bec and bic for possible later use by Trcw and Trwc.`);
l(`When the curve is "new"  it goes through Tate's algorithm to determine the`);
l(`conductor etc., assigns a catalog name, and makes note in the catalog of`);
l(`any twists.`);
l(`  For example if all 28 curves of conductor 448 are introduced (typically`);
l(`Ein is followed by the command Isog which can introduce as many as 8 c\
urves)`);
l(`they are assigned in succession the catalog names A448,..,Z448,A&448,B&\
448..`);
l(`To permute the names, for instance to make them agree with some other`);
l(`list, use Inch. To stop (and restart) curves being entered into the`);
l(`stack and catalog (which may be advisable when large families of`);
l(`curves are generated) use the command Store().`);
    elif i<19 then
l(`ell(args)=NULL`);l(`Ell(args)=NULL`);
l(`   where args can have any one of the following 7 forms`);
l(`   (a1--a6 stands for the sequence a1,a2,a3,a4,a6):`);
l(`             a1--a6             a1--a6,K_`);
l(`            [a1--a6]           [a1--a6],K_         [a1--a6,K_]`); 
l(`              ncur               ncur,K_`);
l(`               cur                cur,K_`);
l(`This is the general input command of the Weierstrass equation `);
x:='x':y:='y':print(y^2+'a1'*x*y+'a3'*y=x^3+'a2'*x^2+'a4'*x+'a6');
l(`regarded as being defined over a field K of characteristic 0, and`);
l(`the optional parameter K_ is a label identifying K, as we will explain.`);
l(`The ai can have any values --- they can be rational, irrational or lit\
eral,`);
l(`but may be restricted when K_ is specified --- see below.`);
l(`ncur is the number of a catalog curve; thus Ell and ell, like Ein and ei\
n,`);
l(`have the facility of calling up curves that are already in the catalog.`);
l(`However any curve introduced by Ell or ell is not put in the catalog, an\
d`);
l(`the curve's name cur has the "generic" value 'cur'. When args=cur or`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`args=cur,K_, then cur can be either a catalog name, and then serves the`);
l(`same purpose as ncur (e.g.,`);
l(`Ell(A11) and Ell(1) are the same), or the "generic" cur='cur', and then`);
l(`Ell will understand this as a handy way of specifying the current values`);
l(`of a1--a6, i.e., a handy way of keeping the same a1--a6 while changin\
g K_.`);
l(`  The field label K_, if absent from args, is assigned the default valu\
e 0;`);
l(`otherwise it can be given any numerical or literal value, including the`);
l(`following which are reserved by apecs to have special meaning:`);
l(`      K_ = 0 :  general (unspecified) field`);
l(`      K_ = 1 :  the rational field Q --- this is reserved for use by ein`);
l(`                (see Menu(ein)), and ell will replace it with K_:=0;`);
l(`      K_ = m, a square-free integer <>1 : the quadratic field Q(sqrt(m))`);
l(`      K_ = T :  the function field Q(T) (the variable must be T)`);
l(`For example, Ell(G15,-5); points apecs to the curve E with Weier.`);
l(`coeff.'s a1--a6=1,1,1,-110,-880 and also points apecs to the field`);
l(`K=Q(sqrt(-5)) whose label is K_=-5. Tor(); now determines the torsion`);
l(`subgroup of E(K) to be of order 4 -- O and the point of order 2 in E(Q)`);
l(`already known to the catalog, plus 2 new points of order 2 defined over`);
l(`Q(sqrt(-5)). E could have been introduced by Ell(17,-5), since G15 is in`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`position ncur=17 in the cat. (confirmed by ein(cur):ncur; ==> 17) or by`);
l(`Ell(1,1,1,-110,-880,-5) or ...`);
l(`Ell(cur,5); keeps the same a1--a6=1--(-880), but shifts apecs's attention`);
l(`to K=Q(sqrt(5)).`);
l(`     For all curves input via Ell, the basic arith. commands`);
l(`Eadd, Sub, Neg, Comb, Ford, .. and certain others On, Subg, Velu, Tw, ..`);
l(`are available, but some, such as Rk, Bas, Isog, .. , are restricted to`);
l(`catalog curves --- those with field label K_=1. Tor works when K_=m_ or T,`);
l(`and whenever all a1--a6 are rational (any K_).`);
l(`     The restrictions on a1--a6 are as follows: (if a1--a6 are not in`);
l(`conformity, K_ will be set to 0)`);
l(`K_ not specified: no restrictions, e.g.`);
print(`Ell(5/2,-(sqrt(2)+log(3))^4,0,A,B^2/C)`);
l(`    and K_ is given the default value 0.`);
l(`K_=0: same as no K_.`);
l(`K_=1: K_ will be reset to 0 (since K_=1 is reserved for exclusive use by`);
l(`      Ein and ein), hence no restrictions on a1--a6.`);
l(`K=a squarefree integer m<> 0 or 1: ell will call qfin(m) (see Menu(qfin))`);
l(`      which sets m_:=m and focuses apecs attention on the quadratic field`);
l(`      Q(sqrt(m_)). a1--a6 must be in Q(w_,sqrt(m_)), where w_ has been`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`      defined by qfin to be an alias for sqrt(m_) or (sqrt(m_)-1)/2,`);
l(`      depending. E.g.,`);
print(`Ell(3/2,0,(2*w_+3/sqrt(5))^6,w_,sqrt(5),5)`);
l(`      Since Ell insists that a1--a6 be algebraic integers in the form`);
l(`      a+b*w_ with a,b in Z, Ell will clear denominators and annnounce`);
l(`      that the actual ai is the input'd ai times uV^i where, in this`);
l(`      example,`);
print(` uV`=10);
l(`      The command Ell(cur,-10); will keep the same a1--a6 but will point`);
l(`      apecs to Q(sqrt(-10)) and w_ will be interpreted as sqrt(-10).`);
l(`      Go(psn) returns to the curve/Q(sqrt(5).`);
l(`      See the Menu entries for Qfin, Fundunit, Minf and Go.`);
l(`K_=T: a1--a6 must be in the rational function field Q(T), and the`);
l(`      variable must be T. As in the previous case, ell will clear`);
l(`      denominators so that a1--a6 are in Z[T].`);
l(`Other K_: no restrictions on a1--16.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`  The user may wish to tag other fields with his own labels K_, but it`);
l(`is only for K_=1,m_,T as above that apecs has special procedures.`);
l(`  In a little more detail, ell does the following:`);
l(`  --- check args`);
l(`  --- store the present curve which is about to be replaced (if there is`);
l(`one) in the stack along with pertinent data, also in the catalog if K_=1`);
l(`(Note: these storages can be toggled off and on; see Menu(ein);)`);
l(`  --- calculate b2,b4,b6,b8,c4,c6,DD,jay`);
l(`  --- if K_ indicates a quadratic field call qfin(K_); (see Menu(qfin);)`);
l(`EXAMPLE: Looking at the output from the following sequence of commands`);
l(`shows apecs engaged in a variety of activities (in particular, quar0`);
l(`calls ell):`);l();
print(`Quar0(1,0,0,3,1);Ein(cur);Bas();Crem();`);
    elif i<21 then
l(`emod(p)=card`);l(`Emod(p)=NULL`);
l(`   where p is a prime and card is the cardinality of the set of points of`);
l(`E mod p including the singular point if p is bad. The variable spur,`);
l(`which is often denoted a_p, is p+1-card; when p is good, spur is`);
l(`the trace of Frobenius and is a Legendre symbol times T[jay mod p]`);
l(`where T is one of the tables froB0,tFr,tFr2 or tFr3 according as p<100,`);
l(`100<p<855, 855<p<1350 or 1350<p<tFr2[1]), with slight additional`);
l(`complications when p<17 or j=0 or 1728 mod p. See also Allp.`);
    elif i<23 then
l(`emods(p1,p2)=NULL`);l(`Emods(p1,p2)=NULL`);
l(`Both commands print a table with column headings p,N,ap,theta,DD,j,--,`);
l(`where p runs through the primes satisfying p1<=p<=p2, N is card as`);
l(`calculated by emod(p),theta is defined by N=p+1-2*sqrt(p)*cos(theta),`);
l(`DD and j are the discriminant and j-invariant mod p, and the last column`);
l(`has the entry "singular-additive", "singular-split multiplicative",`);
l(`"singular-nonsplit multiplicative", "super singular" (i.e. nonsingular and`);
l(`N=1 mod p), or "anomalous" (i.e. nonsingular and N=0 mod p), as appr\
opriate.`);
    elif i<25 then
l(`ford(args)=Qq`);
l(`Ford(args)=NULL`);
l(`   Calculate and list the first t multiples of the point z`);
l(`where args=z,t and z is a point on E in the form x,y or [x,y] or`);
l(`[x,y,n] (the last form can be used only when z has finite order n),`);
l(`and Qq=z,2*z,..,(ord-1)*z where ord=min{t+1,N}, N denoting the order`);
l(`of z, possibly infinity. ord and Qq are global variables that are`);
l(`available after either Ford or ford. If on(z)=false then ford sets ord=0.`);
l(`However this "on" test is disabled if hastype(zi,float) is true; this`);
l(`allows one to get approximate values for the multiples of a real point.`);
    elif i<27 then
l(`gcub(args)=NULL`);l(`Gcub(args)=NULL`);
l(`   where args is a sequence s1,s2,..,sn where n = 10, 12 or 13.`);
l(`These args can be elements of "any" field of char. 0, but`);
l(`n = 10 is allowed only when s1,..,s10 are all rational numbers.`);
l(`s1,..,s10 are the coefficients of the cubic, here called C,`);
  print(`s1*U^3+s2*U^2*V+s3*U*V^2+s4*V^3+s5*U^2+s6*U*V+s7*V^2+s8*U+s9*V+s10=0`);
l(`If nargs=12, s11,s12 are the coordinates U,V of a point on C;`);
l(`if nargs=13 then s13 must be 0 and s11,s12,0 the homogen. coord.'s of a`);
l(`point at infinity (i.e., s1*s11^3+s2*s11^2*s12+s3*s11*s12^2+s4*s12^3=0).`);
l(`If nargs=10 (and si in Q) the procedure searches up to |U|+|V|<=lgcub for`);
l(`an integer point, where lgcub has the default value 10 and can be reset`);
l(`at any time. Clock can be used.`);
l(`   Thus the user has the option of supplying a point on C when s1,..,s10`);
l(`are all rational, but a point must be supplied when they are not all`);
l(`rational.`);
l(`   When a point on C is in hand and the curve is found to be of genus 1,`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`equivalently, non-singular (hence irreducible), a birational`);
l(`transformation to weierstrass form E in which the chosen point is`);
l(`transformed to the point O at infinity is calculated and stored in bic.`);
l(`C is stored in bec (see Trcw and Trwc which use bec and bic).`);
l(`   In the rational case ein is invoked, so the weier. equation is`);
l(`Z-minimal and E is added to the catalog, if not already there. Otherwise`);
l(`Ell is called with field label K_ usually = 0, but if gcub sees that`);
l(`C and the point are defined over the rational function field Q(T), or`);
l(`over the quadratic field Q(sqrt(m_)) (see Menu(Qfin) for m_), then`);
l(`K_ is set to T, m_ respectively.`);
l(`   The birational transf. is a composition of a varying`);
l(`number of changes of variables. The subprocedures unk and xnk calculate`);
l(`(formally) the composition of functions and inverse functions respec\
tively.`);
l(`   EXAMPLE: John McKay wanted a quick way to calculate the j-invariant`);
l(`of various E/Q(T), the simplest example being`);
    print(U^3+V^3+T^2*U*V+1=0);
l(`A convenient answer is given by`);
    print(` Gcub(1,0,0,1,0,T^2,0,0,0,1,0,-1);factor(jay);`);
l(`Noticing that [+-T,-1] are points on C, one can follow this up with,`);
l(`e.g., Trcw(T,-1);Trcw(-T,1);`);
    elif i<28 then 
l(`Genj(args)=NULL      where args=j or j,k; j is required, k is optional`);
l(`  If absent from args, k is set=1;`);
l(`  if j=0 call Ein(A27,k) (see Menu(ell) for purpose of k);`);
l(`  if j=1728 call Ein(A32,k);`);
l(`  otherwise put a1=1, a2=a3=0, a4=-36/(j-1728), a6=-1/(j-1728) and then`);
l(`  if j is rational call Ein(a1,..,a6,k), otherwise Ell(a1,..,a6,k)`);
    elif i<29 then 
l(`genj(args)=curve is, cur  #if j is rational`);
l(`          = yo  #a list of the 5 Weierstrass coef.'s if j not in Q`);
l(`      where args=j or j,k; j is required, k is optional`);
l(`  If absent from args, k is set=1;`);
l(`  if j=0 call ein(A27,k)   (see Menu(ell) for purpose of k)`);
l(`  if j=1728 call ein(A32,k);`);
l(`  otherwise put a1=1, a2=a3=0, a4=-36/(j-1728), a6=-1/(j-1728) and then`);
l(`  if j is rational invoke ein(a1,..,a6,k), otherwise ell(a1,..,a6,k)`);
    elif i<31 then
l(`ht(args)=ght`);l(`Ht(args)=NULL`);
l(`   where args is a point on E in the form x,y or [x,y], or even just x --`);
 l(`after all, the canon. ht. depends only on x -- and ght is the`);
l(`point's (global) canonical height as calculated by Silverman's modification`);
l(`of Tate's algorithm to "Digits" digits of accuracy. The maple variable`);
l(`"Digits" can be reset at any time in apecs (default=20).`);
l(`Note: sw = number of switches (b<-->1-b) used in Silverman's algorithm.`);
    elif i<33 then
l(`sub(z1,z2)=z3        Sub(z1,z2)=NULL`);
l(`   calculate z3=z1-z2 where zi=[xi,yi] or [xi,yi,ni] are points on E`);
l(`Like Eadd and eadd, Sub and sub work for "general" points.`); 
    elif i<35 then
l(`isog()=nisog,disog`);l(`Isog()=NULL`);
l(`   where nisog is a list of the catalog numbers of all isogenous curves`);
l(`(this command enters into the catalog any of these curves that are not`);
l(`already there) and disog is a list of the corresponding degrees.`);
l(`Note the convention: a degree is listed as negative when the kernel`);
l(`contains irrational points; of course the kernel as a whole is rationally`);
l(`defined. For visog see below.`);
l(`Find all isogenous curves and set the flag qR=0. For later, higher values`);
l(`of qR see Bas.`);
l(`If there is any twist whose isogeny network is already worked out then`);
l(`twist that network to give the network of the present curve. Otherwise`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`check for the existence of isogenies`);
l(`   (i) in the cases where the modular curve has genus 0 see if the Fricke`);
l(`polynomial has a rational root (these polynomials are listed by the`);
l(`command Nota), using common sense to cut the amount of checking; e.g.,`);
l(`since there are no non-cuspidal rational points on the mod. curve of`);
l(`level 35, E can't have both an isogeny of degree 5 and another of degree 7`);
l(`or if E is semistable then it can't have an isog. of degree 13, by a`);
l(`theorem of Serre;`);
l(`   (ii) in the cases of higher genus see if jay is in the list ljay of`);
l(`of special values - this includes the higher CM cases, although they`);
l(`will have already been spotted by the twist check - unless the examples`);
l(`of curves with the conductors 11^2,..,163^2 have been forcibly removed`);
l(`from the catalog (this is not recommended).`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   In any case calculate E/each finite cyclic subroup G by formulas`);
l(`derived from Velu's and store the details in visog. In apecs versions<2.5`);
l(`these details were restricted to isogenies with G consisting of rational`);
l(`points --- then the entry in visog was a coded list of these points.`);
l(`(This also involved entering info. in the catalog variable tuQ).`);
l(`Otherwise the visog entry was left blank (except for certain cases |G|=\
3..)`);
l(`Now the entry has the form [k,x,y] (and tuQ is no longer used) where`);
l(`k is a poly. in X of degree trunc(|G|/2) and whose roots are the `);
l(`unduplicated X-coordinates of the non-zero points in G; x and y are`); 
l(`rational functions in X and Y --- the coordinate transformations from E`);
l(`to E/G. (Old style entries of visog are upgraded as the occasion arises.)`);
l(`visog is used by Tri (see Menu(tri)) and Velu (also internally by apecs`);
l(`e.g. when the visog entry is in place by apq to improve the list of`);
l(`independent points of infinite order on E/G when such are found on E.)`);
    elif i<36 then
l(`If xxx represents an apecs command then the command Menu(xxx) first prints`);
l(`the value of xxx. For example, Menu(isog) first says that`);
l(`isog()=nisog,disog which means that a command of the form var:=isog();`);
l(`will assign the value nisog,disog to the variable var (the name var is not`);
l(`used by apecs and so is available; nisog and disog are apecs variables`);
l(`whose meaning is explained by the apecs command Nota();). Menu(xxx) then`);
l(`describes the parameters, if any, required by xxx, then further details.`);
    elif i<40 then
l(`mulp(args)=Nz`);l(`Mulp(args)=NULL`);
l(`mult(args)=Nz`);l(`Mult(args)=NULL`);
l(`   where args=N,z where N is in Z and z is a point on E, or on E mod p`);
l(`in the cases Mulp,mulp (the value of p must be preset to a prime), and`);
l(`Nz=[xm,ym] is N*z on E in the cases Mult,mult and on E mod p in the cases`);
l(`Mulp,mulp. Notice that mulp and mult (but not Mulp, Mult) can be`);
l(`concatenated with other commands e.g., Trwc(mult(-3,xx)).`);
l(`These commands work "in general" e.g. with a curve introduced by ell,`);
l(`except Mulp and mulp require the Weier. coeff.'s at least to be rational`);
l(`with denominators prime to p.`);
l(`Like Eadd and eadd, Mult and mult accept points with irrational coord.'s`);
l(`also floating point coord.'s for approximate calc.'s`);
    elif i<42 then
l(`om()=Omega`);l(`Om()=NULL`);
l(`   The present curve must have all Weierstrass coeff.'s a1,..,a6 real`);
l(`(rational or float), and so isn't necess. a catalog curve intro. by ein.`);
l(`and with discriminant DD<>0. The following are calculated:`);
l(`The lattice periods omega[1],omega[2] (omega is a table) where omega[1]`);
l(`is real and positive, and omega[2] is such that omega[2]/omega[1] has`);
l(`positive imaginary part and real part=0 (when DD>0) or -1/2 (when DD<0).`);
l(`In the former case Omega (the factor in the Birch, Swinnerton-Dyer conj.)`);
l(`= 2*omega[1], and in the latter case Omega=omega[1]. Set`);
l(`omegA:=[[Omega,omega,Digits],..] where ,.. represents the sequence`);
l(`(usually empty) of p-adic periods calc. by omp, each a list of length>3.`);
l(`Thus nops(omegA)>0 and nops(omegA[1])=3 signal that om has been called`);
l(`previously.`);
l(`   If Digits is set to a higher value than that stored in omegA, and`);
l(`om is called again then the data will be recalculated to the higher`);
l(`accuracy. For omegA[2],.. see Menu(omp).`);
    elif i<44 then
l(`omp(args)=NULL`);l(`Omp(args)=NULL`);
l(`   where args is either n or p,n; the absence of p in the first case`);
l(`means do omp(p,n) for all relevant p, namely those primes that occur in`);
l(`the denominator of jay: calculate the "p-adic periods" (a misnomer) and`);
l(`store succesively in positions 2,3,.. in the list omegA. (For omegA[1]`);
l(`see Menu(om).) For such p the curve E is isomorphic over the p-adic field`);
l(`Q_p(sqrt(-c6)) to Tate's curve Gm/q^Z and denoting by u the ratio of their`);
l(`canonical differentials, as explained by Henniart and Mestre in their 1989`);
l(`C.R.Acad.Sci. Paris paper, q and u^2 are in Q_p^*. These procedures calc\
ulate`);
l(`q and u^2 to n p-adic digits and store the results in the first available`);
l(`omegA[i] in the format p,n,[c,d0,d1,..,dn],[e,f0,f1,..,fn] where`);
l(`q=p^c*(d0+d1*p+..+dn*p^n), u^2=p^e*(f0+..).`);
l(`#### haven't yet got around to programming p=2 and additive type I* ####`);
    elif i<48 then
l(`quar(args)=(one of) -2,-1,0,P__,[],[0],[S6,S7],or FAIL -- see table below`);
l(`Quar(args)=NULL`);
l(`quar0(args)= same as quar`);l(`Quar0(args)=NULL`);
l(`   where args is a sequence S1,S2,.. of 5, 6 or 7 arguments.`);
l(`S1,..,S5 are the coefficients in V^2=S1*U^4+S2*U^3+S3*U^2+S4*U+S5;`);
l(`if both S6,S7 are present, they are the coordinates of a point [U,V]`);
l(`on the curve.`);
l(`If S6 is present, but not S7, where S6 can have any value (e.g.  5 or X)`);
l(`then the point at infinity is taken as the "rational" point on the`);
l(`quartic, i.e., the point [0,sqrt(S1)] on (V/U^2)^2=S5*(1/U)^4+..+S1.`);
l(`Quar0 is same as Quar except it treats rational Si as irrat. -- see below.`);
l(`    If S6 is not present the procedure attempts to supply a point,`);
l(`as follows.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    If S1..S5 are rational, it first checks whether S5 is a square;`);
l(`if so, it takes the point [0,sqrt(S5)]. Failing that, it asks: are the`);
l(`places at infinity rational, i.e. is S1 a square. If so it proceeds`);
l(`appropriately. Next, if the quartic has a rational root r then it takes`);
l(`[S6,S7]=[r,0]. Otherwise, at this point if one of S1,..,S5 has denom.>1,`);
l(`then the denom.'s of S1,..,S5 are cleared by a suitable substitution`);
l(`V/uV for V and quar is re-called with new, integral S1,..,S5. It then`);
l(`tests the quartic for local solvability. If it is not solvable over some`);
l(`rational P__-adic field, then P__ is returned by quar; or if not solv.`);
l(`over the reals then P__=infinity is returned. If the quartic survives,`);
l(`quar searches for [U,V] with numer(U)+denom(U)<=lquar+1 where lquar=100`);
l(`and can be reset at any time; e.g., lquar:=200; prior to calling quar.`);
l(`Also, clock can be used.`);
l(`    If some S1..S5 is irrat. and no point [S6,S7] is supplied by the \
user,`);
l(`then quar supplies [S6,S7]=[0,sqrt(S5)]. Quar0 acts the same way even`);
l(`when all S1..S5 are rational.`);
l(`    In any case the procedure checks that the curve is irreducible of`);
l(`genus 1 and then finds a birationally equiv. Weierstrass equation in`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`the variables X,Y. If the Si are rational it then finds the min. Weier.`);
l(`equation. (To inhibit the transformation to minimal Weier. when the Si`);
l(`are rational, use Quar0 instead of Quar.) As in the case of Gcub, the`);
l(`original U-V curve is stored in bec and the transformations between it`);
l(`and the X-Y Weier. curve are stored in bic --- see Menu(Gcub) for more`);
l(`details. Also see Menu(Trcw) for transfering points between these curves.`);
l(`The value of quar has the following meaning:`);
l(`  -2     wrong args`);
l(`  -1     [S6,S7] not on curve`);
l(`   0     discriminant of quartic DD_=0`);
l(` P__     quartic not locally sol. at P__ = infinity or a rational prime`);
l(`  []     point at infinity taken as O --- called quar(S5,..,S1,0,sqrt(S1))`):
l(` [0]     S1=0 so called ein0 or ell`);
l(`[S6,S7]  =point taken as O`);
l(` FAIL    couldn't find rational point, or timed out by clock`);
l(`         --- increase lquar and/or don't use clock`);
l();
l(`Thus if quar (or quar0) returns a list then the quartic is "elliptic",`);
l(`but if it returns FAIL then the nature of the quartic is undetermined.`);
    elif i<50 then
l(`rk(args)=NULL      N.B. when the conductor Nc is large it may be`);
l(`Rk(args)=NULL      advisable to use Rk0, Rk1 or RkNC instead (qv.).`);
l(`   args is optional; if present, rk sets dil:=args, otherwise dil:=30`);
l(`Usually args is absent --- one usually calls Rk(); see below.`);
l(`   Calculate r4=the rank, rc=the "quality index" of r4, and r1=`);
l(`a conjecture-free upper bound for the rank.`);
l(`   N.B. the value of r4 is reliable only when rc>=0; otherwise rc=-1`);
l(`and r4 is either -1 (its default) or an upper bound for the rank which`);
l(`is conjectural unless r4=r1`);
l(`   Note that MM=card(RR) is always a lower bound for the rank: RR is a`);
l(`list of points that are lin. indep. mod torsion. (Possib. better lower`);
l(`bound: MN=max{card(RR of E') : E' isog. to E}.) RR can be augmented`);
l(`independently of Rk by the command Seek which looks for points, or`);
l(`by Tri which transfers points between isogenous curves, or by`);
l(`App which appends points known from some other source.`);
l(`   Whenever apecs finds a new point P, whether in Rk, Seek,...,`);
l(`either P is added to RR (when independent of the points already in RR),`);
l(`or P replaces some point (if that gives a smaller grammian -- height`);
l(`pairing -- determinant) or P is added to the list uR of redundant points.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   r4 (when>=0) in general has been determined with the aid of conjectures`);
l(`depending on the value of rc, as we explain in a moment. On the other`);
l(`hand, r1 is a firm upper bound for the rank, independent of any conj.`);
l(`unless it has its default value of -1`);
l(`   The value of rc indicates the "quality" of r4:`);
l(`T stands for the Taniyama conj.; T-p means T and parity of r4`);
l(`as determined by a root number calc. by the procedure roha() or (inadvisabl\
y)`);
l(`analytically from the sign of the functional equation by the proc. sfe().`);
l(`B-SD stands for the Birch, Swinnerton-Dyer conjecture and R.H.for the`);
l(`Riemann Hypothesis: all zeros s+it of L(s) with 1<=s<=2 have s=1.`);
l(`    In the following table the values of rc with an asterisk are`);
l(`obsolescent: they persist in earlier entries in the catalog, but since`);
l(`version 5.5, the trio rc,r4,r1 is treated equitably among all members of`);
l(`the isogeny class --- rk calls isog and then (and later when there is`);
l(`any change in the trio) for all the curves updates the cat. entries to`);
l(`their best currently available values. Thus once rk is called (either`);
l(`directly or indirectly by one of bas,rk0,rk1,rkNC,Rk,Bas..) rc will not`);
l(`be any one of 0,1,4,6, and will be the same for all curves isogenous to`);
l(`the present curve --- referred to as "new rc" in the table.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`     |new|`);
l(`  rc |rc | provenance of r4`);
l(`  -1 |-1 | no firm value yet for the rank; if r4>0 it is an upper bound`);
l(`     |   | for the rank which is conjectural unless r4=r1`);
l(`  *0 | 2 | from r4 of isogenous curve which used (at least some of)`);
l(`     |   | T, B-SD, R.H.`);
l(`  *1 | 3 | from r4 of isogenous curve which used only T-p`);
l(`   2 | 2 | used (at least some of) T, B-SD, R.H.`);
l(`   3 | 3 | used only T-p`);
l(`  *4 | 5 | from r4 of isogenous curve using no conjectures`);
l(`   5 | 5 | used no conjectures`);
l(`  *6 | 7 | from an isogenous curve for which MM (the cardinality of the`);
l(`     |   | list RR of independent points) = r1.`);
l(`     |   | Thus r4 is the conjecture-free value of the rank.`);
l(`   7 | 7 | as in *6: MM=r1 either for the present curve or an isog. one`); 
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    Rk tries to get r4 using as few conjectures as possible.`);
l(`The procedure terminates as soon as rc>=0 (or rc>=a higher value if rk`);
l(`has been called by Rk1, Rk0 or RkNC --- see Menu(Rk1) etc.); thus not`);
l(`all the following steps are necessarily carried out. When called by`);
l(`Rk1, Rk0 or RkNC, the proc. may terminate to report failure: rc=-1.`);
l(`In principal Rk itself will aways succeed, but in practice it may bog`);
l(`down in the final L-series step.`);
l(`    Let E denote the present curve and let C denote the set of all E'`);
l(`(rationally) isogenous to E, including E'=E.`);
l(`    Step 1. Find the best (smallest) value for r1 among Mazur's upper`);
l(`bounds (description below) for all E' in C to which Mazur applies.`);
l(`E.g: Rk applied to the example Exam(3,124,67) (see Menu(Exam)) gets`);
l(`r1=0 and so rk exits with r4=0 and rc=7. Of course usually C consists of`);
l(`E alone, so no Mazur, and this step leaves r1=-1.`);
l(`    Step 2. This step is also applied to all E' in C. If NN=card(torsion)`);
l(`is odd then seek(dil) is applied, i.e., search for points on E' up to`);
l(`naive height dil; usually rk is called without args, so dil has its`);
l(`default value 30, and the search is modest --- so as not to miss any`);
l(`obvious points.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    In NN is even, then 2-descent, including 2-nd descent when necess.`);
l(`(see Elliptic curve Handbook, 3.6.6, available by anon. ftp from same`);
l(`site as apecs) is performed on each E'. Of course if a particular`);
l(`2-descent E'--->E'' is performed then E''--->E' is skipped over.`);
l(`E.g. Rusin's examples Ein(0,0,0,-34225,0) and Ein(0,0,0,-(29*61)^2,0)`);
l(`find r4=0, and nicely show the importance of not just looking at a`);
l(`single E--->E'. Sometimes even looking at all 2-descents out of E is`);
l(`not enough, and other E'--->E'' are considered. However, if at the end`);
l(`of all this rk has not reached a conclusion, then the uncertain quartics`);
l(`(homogeneous spaces or torsors) are searched for points "up to dil". `);
l(`These quartics are stored (with other info) in the list dolt.`);
l(`The latest (completed) search limit dil is stored as dill.`);
l(`     Step 2 bis. The following example illustates repeated calls to`);
l(`rk in the case when NN is even and rk leaves dolt unempty; it is`);
l(`necessary to use RkNC in the subsequent calls since a second call to`);
l(`rk itself would merely produce "already found r4=1 with rc=3".`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`         Exam(6,78);Rk();  ===>  r4=1, rc=3 (hence RR=[], MM=0)`);
l(`         RkNC(50);  ===>  nada, i.e. continuing the search up to dil=50`);
l(`                          on the quartics in dolt finds no points`);
l(`         RkNc(100);  ===> finds a point on one of the quartics, hence`);
l(`                          MM=1 (check out RR), r4=1 with rc=7.`);
l(`     Step 3. (permitted by Rk, Rk0, and Rk1, but not by RkNC):`);
l(`roha() is invoked to find the root number SFE (which stands`);
l(`for Sign of the Functional Equation) hence the parity of the rank`);
l(`conjecturally -- (-1)^r4=SFE -- to be used if needed.`);
l(`     Step 4. (permitted by Rk and Rk1 but not by Rk0 and RkNC):`);
l(`Mestre's upper bound for the rank is calculated by mest(args), followed by`);
l(`seek(args); these proc.'s are repeated up to 4 times with successively`);
l(`higher args. Mestre uses the analog of Weil's exact formula which`);
l(`necessitates assuming T, B-SD and R.H.`); 
l(`After an inconclusive run of Rk1, one can call (repeatedly)`);
l(`Mest(args) with args > the current value of mstR.`);
l(`     Step 5. (not permitted by Rk0, Rk1 or RkNC): As a "last resort"`);
l(`rank is determined as the smallest r such that L^{r}(1)<>0.`);
l(`(See Menu(FnL)). For "large" E this step can take forever.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`A few more details :---`);l();
l(`  Mazur's upper bound:`);
l(`     Simplifying slightly results of [B. Mazur, Rational points of`);
l(`abelian varieties with values in towers of number fields, Invent.`);
l(`Math. 18(1972), 183--266], if p is a prime where E is good or`);
l(`mult. and E(Q) is divisible by Z/pZ, i.e. E(Q) contains a point of order p`);
l(`with integral Weierstrass coord.'s (which is automatic when p>2)`);
l(`or divisible by mu_2, i.e. there is a point of order 2 with coordinates`);
l(`of the form [a/4,b/8], a and b odd integers)`);
l(`then a value for r1 is a+m-1 which can be improved to a+m-1-u-e`);
l(`when E(Q) is div. by Z/pZ, where:`);
l(`a=the number of additive primes;`);
l(`m=the total number of bad primes (not just the number of mult. ones);`);
l(`u is the number of primes s such that`);
l(`  (i) E has mult. red. at s;`);
l(`  (ii) p does not divide the exponent of s in DD; and`);
l(`  (iii) s is not 1 mod p;`);
l(`e=0 except when p=2 and there is a prime s satisfying (i), (ii) and`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`  (iii') s is not 1 mod 4`);
l(`and then e=1.`);
l(`If there is more than one such p, the min. upper bound is taken for r1.`);
l();l(`2-descent:`);
l(`  If E has a point of order 2 the usual simple 2-descent is used. The`);
l(`homogeneous spaces are sieved by Hensel (preceeded by a quad. norm res. t\
est).`);
l(`For the homog. spaces A*U^4+B*U^2*V^2+C*V^4=W^2 that remain, 2nd \
2-descent`);
l(`is performed. The descendant quartics are Hensel sieved, and those that`);
l(`survive are stored along with the original homog. spaces in dolt.`);
l(`Then all the quartics in dolt are searched for rational points up to`);
l(`naive height dil. The default value of dil is 30; it can be given a`);
l(`different value by using args: Rk(args) (or rk(args)) sets dil=args;`);
l(`dil will revert to its value 30 next time rk is called --- unless args`);
l(`again specifies another value. If rk has already found a value for r4`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`with rc>=0, use RkNC(args) instead of Rk(args) to extend the search`);
l(`on dolt.`);
l(`  Any points found on a quartic are transformed into points on E and`);
l(`stored (if not "redundant") in the list RR. Any time a decision about`);
l(`the rank can be made, the search is called off.`);
l(`  If rk deems it appropriate,`);
l(`a direct search for points on E up to naive height dil is made.`);
l(`  Depending on circumstances, not all the isogenous E' have their RR`);
l(`necessarily updated following improvements in the RR of E: the primary`);
l(`mandate of rk is to determine r4. In such cases one can use Tri, qv.`);
    elif i<52 then
l(`rk0(args)="rank = r4" or "rank not determined"`);
l(`Rk0(args)=NULL`);
l(`  invoke rk(args) with rflag=2 which means disallow use of the procedures`);
l(`mest(); and fnL(); I.e., no conjectures are used except possib.`);
l(`parity of rank = that determined by roha(); To disallow even that,`);
l(`use RkNC(); (q.v.)`);
    elif i<54 then
l(`seek(args)=NULL`);
l(`Seek(args)=NULL               See also Seek1 and Search.`);
l(`  These proc.'s look for points on E defined over Q or over Q(T):`);
l(`--When E is defined over Q(T) (with K_=T) then args is not used (take`);
l(`args=NULL) and seek is a one-shot deal: a modest search is made, and a`);
l(`second call to seek will give "already done".`);
l(`--When E is defined over Q, either by Ein (so K_=1) or Ell (so K_<>1),`);
l(`then args is either l (a positive integer) or l,seekn (both positive`);
l(`integers).`);
l(`  Assuming for the moment that seekn is absent, E is  searched for`);
l(`rational points [x,y] where the naive height max{|numer(x)|,denom(x)}`);
l(`ranges between sB+f and l+f, then set sB:=l, where f is 0 or an adjust-`);
l(`ment because of the location of the real point(s) [x_i,y_i] of order 2.`);
l(`Thus when seek is called again it remembers where it left off.`);
l(`  When seekn is given a positive integer value by a 2nd args, seek`);
l(`proceeds as usual but is terminated as soon as RR (="best so far known"`);
l(`list of points that are independent mod torsion) contains seekn points,`);
l(`and sB is given the value approriate to the cut off point. For example,`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`when rk gives positive values for r4 and rc (this would be for an E with`);
l(`K_=1; see Menu(rk)), the call Seek(10^6,r4) might find r4 independent`);
l(`points quickly, and not "waste time" searching further. An example is`);
l(`given in Menu(Heeg).`);
l(`IN MORE DETAIL:`);
l(` --- calc. the sequence PP of torsion points, if not already done;`);
l(` --- when e:=max{x_i}>0, also look for integral points with e<x<=e+args;`);
l(` --- process any points found by the procedure ap1_ as follows.`);
l(`     (RR is current list of independent points mod torsion; uR is the`);
l(`     set of points +-P+T where P is in RR and T is a torsion point;`);
l(`     yR is (after ap1_) a list of points that are rationally but not`);
l(`     integrally dependent on RR, and thus must be empty when RR is a`);
l(`     basis);`);
l(`  -- adjoin new point Rz to the list yR (usually this is temporary);`);
l(`  -- loop thro' yR: can RR be improved? Repeat 'til RR stabilizes;`);
l(`  -- carry any changes over to isogenous curves if feasible;`);
l(`  -- put Rz and "derived" points in uR.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(` *** To do a straight search with no "f" adjustment and no "ap1_"`);
l(`     processing see the procedure Search.  To search for integral points`);
l(`     only use Seek1.`);
l(`The case E/Q with K_<>1 (usually K_=0, the default value given by Ell)`);
l(`     can be useful, e.g. when a1,a2,a3 are large and a4=a6=0. The points`);
l(`     found are stored in the list searchl, besides the usual "RR and uR`);
l(`     work". Note that RR is not lost when E is converted to another K_,`);
l(`     as the following example illustrates.`);
l(`          Ell(0,0,7,0,0);`);
l(`The curve is y^2+7y=x^3 with K_=0; in this discussion let's call it C.`);
l(`          Seek(10);`);
l(`first calculates PP=[0,0,3],[0,-7,3], so the order of the tor. subgroup`);
l(`is NN=3. Then it discovers [0,0],[2,1],[-7/4,-7/8], notes the first is a`);
l(`torsion pt., sets RR=[[2,1]], and finds that the third is redundant,`);
l(`i.e. of the form n*[2,1]+T, T a tor. pt. To determine n and T we cannot`);
l(`call Expr directly (because the canonical height function ht works only`);
l(`for Z-minimal E..). Instead we invoke`);
l(`          Ein(cur);`);
l(`This calls Ein(0,0,7,0,0) which produces the Z-minimal curve`);
l(`y^2+y=x^3+12 --- let us call it W. W is now the "present curve" and both`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`PP and RR have been transformed from C to W: now PP=[0,3,3],[0,-4,3],`);
l(`and RR=[[2,4]]. (If desired, Go(psn) would return to C and of course`);
l(`PP,RR (and uR..) would be restored to the correct values.) Let us define`);
l(`          P1:=trcw(-7/4,-7/8);`);
l(`This sets P1=[-7/4,17/8], the transf. of [-7/4,-7/8] in the birational`);
l(`map C-->W. (Note that trcw and trwc are available only when present`);
l(`curve is W. See Menu(trwc) for more detail.) Now`);
l(`          Expr(P1);===>`);
l(`          [-7/4,17/8]=-[2,4]+[0,-4]  ([0,-4] is a torsion point)`);
l(`Since the birat. maps C-->W and W-->C map O to O, they are elliptic`);
l(`curve morphisms and so preserve + and -. Thus, since trwc(0,-4)=`);
l(`[0,-7], etc., we deduce that on C`);
l(`          [-7/4,-7/8]=-[2,1]+[0,-7]`);
l(`which of course can be verified, e.g. by`);
l(`          Go(psn); Eadd([-7/4,-7/8],[2,1]);`);
l(`    Final note: If all that is wanted is a set of independent points,`);
l(`say to confirm a value of the rank, or to obtain a lower bound, then`);
l(`provided there are sufficiently many integral points Seek1 can be much`);
l(`quicker. A good example of this is Ein(623); Seek1(50000) quickly finds`);
l(`10 independent points, while Seek(50000) grinds on for a long time.`);
    elif i<56 then
l(`sfe()=SFE`);l(`Sfe()=NULL`);
l(`Optional argument: hyy --- use at most hyy terms in the series.`); 
l(`  Calculate the sign SFE of the functional equation of the cusp form f(t)=`);
l(`sum(an*exp(2*pi*i*n*t)), assuming the Taniyama conjecture, as the sign`);
l(`of f(2*i/r)/f(i/(2*r)) where r=sqrt(Nc)=the square root of the conductor`);
l(`the series being calculated to about 4 digits of accuracy (usually to`);
l(`about 1*r or 2*r terms). Also compare with the parity of the rank if that`);
l(`has already been determined by other means (now assuming also the Birch,`);
l(`Swinnerton-Dyer conjecture), and transfer the value of SFE to all`);
l(`isogenous curves`);
l(` Clock can be used.`);
l(`NOTE: the procedures Roha, roha are much quicker and not dependent on`);
l(`approx. sums of infinite series.`);
    elif i<58 then
l(`sha(args)=NULL`);l(`Sha(args)=NULL       ---  usually args is absent`);
l(`  Calculate the order shaf of the Shafarevich-Tate group, assuming the`);
l(`Birch, Swinnerton-Dyer conjecture. If args is absent, Bas must be done`);
l(`beforehand so that RR is a Mordell-Weil basis, well at least a "probable`);
l(`basis", i.e., qR>=3; see Menu(bas) for details about qR. If possible, the`);
l(`value of shaf will be inferred from that of an isogenous curve, otherwise`);
l(`r4-th derivative of the L-function is calc. to appropriate accuracy.`);
l(`    However if args<>NULL`);
l(`then, provided that the sign of the functional equation SFE is known i.e.`);
l(`SFE has one of the values 1,-1, and not its default value 0, and this \
value`);
l(`is in agreement with the parity of the number of points`);
l(`in RR, then it is assumed that RR is a basis and shaf is calculated.`);
l(`   Clock can be used.`);
    elif i<62 then
l(`trcw(args)=p, p_at_inf_w[n] or NULL     trwc(args)=p or NULL`);
l(`Trcw(args)=NULL                 Trwc(args)=NULL`);
l(`Let C be curve #n listed in bec (see below for details).`);
l(`  Trwc(z,n) transfers point z on E to point p on C (w is for Weierstrass)`);
l(`  Trcw(z,n) transfers point z on C to point p on E`);      
l(`  Trwc(n) and Trcw(n) both display C and the transformations between C a\
nd E`);
l(`In any of the above n may be absent -- then it is given the value 1.`);
l(`z has one of the forms`);
l(`     [x,y] or just x,y (an affine point);`);
l(`     [x,y,0] (a point at infinity in homogeneous coord.s);`);
l(`     [] which stands for the set of point(s) at infinity ---`);
l(`apecs understands: E has a unique point at infinity, often denoted O;`); 
l(`a cubic introduced by Gcub has 3 (perhaps with repetition); a quartic`); 
l(`introduced by Quar has two places at infinity, both centered at the`);
l(`same point. We will explain trcw([]) and trwc([]) in a moment.`);
l(`     x and y are not required to be rational; they will typically be`);
l(`irrational in the case of a curve with irrat. coeff.'s introduced by`);
l(`Quar or Gcub.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    bec is a list of polynomials in the variables U,V; the equation`);
l(`bec[n]=0 describes a curve C birationally equiv. to E, whose Weier.`);
l(`equation has X,Y as variables. bic[n] is a list whose entries are`);
l(`8 polynomials that describe the birat. transform.'s, and the point`);
l(`on C that corresponds to O on E; the first 4 are poly.'s in X,Y`);
l(`and the second 4 in U,V.`);
l(`The birat. transf's are: U=B1/B2, V=B3/B4, X=B5/B6, Y=B7/B8 where Bi`);
l(`denotes (in this discussion only) bic[n][i].`);
l(`    Entries in bec, bic are generated by the commands Quar, Gcub, Genj,`);
l(`Ein0 and also may be generated "internally", e.g. a call to Rk may`);
l(`result in one or more entries.`);
l(`    Any new (nonredundant) points of infinite order that show up on E`);
l(`are saved in RR. Note that bec and bic are available only  during the`);
l(`present apecs session - they are saved only in the stack, not the cat.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    When trcw([]) or trwc([]) is called, apecs creates two lists, which`);
l(`for convenience we call A and B. A is the list of points at infinity`);
l(`on C (in the case of the quartic, A=[[0,1,0],[0,1,0]], representing`);
l(`the two places at inf.), and B is the list of the images of the points`);
l(`in A under the birat. trans. A and B become the n-th entries in (the`);
l(`list of lists) p_at_inf_c and p_at_inf_w respectively. Trcw([],n)`);
l(`displays all of A, while if [x,y,0]=A[i] then Trcw(x,y,0,n) displays`);
l(`B[i]; Trwc([],n) displays bic[n][9].`);
l(`Example: Trwc(mult(-2,RR[3])); will find the image p in C #1 of`);
l(`-2 times point no. 3 of E's list RR.`);
    elif i<64 then
l(`tw(a)=[a1,..,a6], or the statement curve is cur`);l(`Tw(a)=NULL`);
l(`  where cur is the new curve obtained by twisting the present curve by a:`);
l(`  cur becomes the present curve, and its Weier. coeff.'s are a1,.. .`); 
l(`  ** If tw or Tw is called when the present curve is a catalog curve`);
l(`  (so that cur = the curve's cat. name, such as A11, and the base field`);
l(`  is Q, so the code K_=1 -- see Menu(Ell)), then tw calls`);
l(`                  ein(0,0,0,-27*c4*a^2,-54*c6*a^3);`);
l(`  Note that since the twist becomes the present curve, for example the`);
l(`  sequence Tw(-3/7);Tw(-14); ends up with the same present curve as Tw(6)\
;`);
l(`  In this case tw evaluates to "curve is",cur.`);
l(`  ** If tw or Tw is called when the present curve is not a cat. curve`);
l(`  (so that cur='cur' -- cur is unevaluated, equivalently, K_<>1), or`);
l(`  if a is not rational, then tw calls`);
l(`      ell(a1,sex_(a*a2+(a-1)*a1^2/4),a3,sex_(a^2*a4+(a^2-1)*a1*a3/2),`);
l(`          sex_(a^3*a6+(a^3-1)*a3^2/4),K_)`);
l(`  where sex_ is defined in apecs by`);
l(`          sex_:=proc(z) simplify(expand(z)):end:`);
l(`  Re the above args. in ell, see Elliptic Curve Handbook, p.410.`);
l(`  tw evaluates to [a1,..], a list of the new present curve's coeff.'s.`);
    elif i<66 then
l(`fnL(args)=L^{r}(1)`);l(`FnL(args)=NULL`);
l(`  where args=r`);
l(`  or    args=r,de`);
l(`  or    args=r,de,hyy`);
l(`Estimate the r-th derivative of the L function at s=1.`);
l(`  For technical reasons the parity of r must be the same as that of`);
l(`SFE (the sign of the functional equation); if SFE is not known FnL will`);
l(`call Roha first.`);
l(`  The optional parameter de must be a pos. integer: it specifies that`);
l(`L^{r}(1) is to be calc. to within +-10^(-de). Default de=4.`);
l(`  The optional parameter hyy must also be a pos. integer, and the param\
. de`);
l(`must be present: it specifies that L^{r} is to be calc. using at most hyy`);
l(`terms in the series. Of course the accuracy indicated by de is only assure\
d`); 
l(`when the calc. is terminated with fewer than hyy terms. The purpose of hyy`);
l(`is to limit the length of the calc. in difficult cases; FnL can be called`);
l(`again with a larger hyy.`);

    elif i<68 then
l(`mest(args)=Mestre's upper bound for the rank`);
l(`Mest(args)=NULL`);
l(`   This uses the analog of Weil's exact formula for L(s) and so depends on`);
l(`assuming T, B-SD and R.H. Used by rk if it can't determine the rank`);
l(`without these conjectures. Series is summed for prime powers<args.`);
l(`When args is NULL, the result of the previous call to mest is stated:`);
l(`the value mstr of the parameter and the value mstR of the upper bound`);
l(`obtained. Note mest(XX) can be forced to calculate when XX<mestr by`);
l(`pre-setting mestr:=0.`);
    elif i<70 then
l(`The apecs commands Trf,trf have been deleted.`);
    elif i<72 then 
l(`on(z)=true or false`);
l(`On(z)=NULL`);
l(`   Tests whether the point z (in any of the forms x,y [x,y] [x,y,n])`);
l(`   is on E.`);
    elif i<74 then
l(`onp(z)=true or false`);
l(`Onp(z)=NULL`);
l(`   Tests whether the point z (in any of the forms x,y [x,y] [x,y,n])`);
l(`   is on E mod p. The prime p must be preset.`);
    elif i<76 then
l(`rk1(args)="rank = r4" or "rank not determined"`);
l(`Rk1(args)=NULL`);
l(`  invoke rk(args) with rflag=1 which means go as far as Mestre's`);
l(`upper bound if necess., then return (hence don't calc.`);
l(`any L^{r}(1): THIS MAKES Rk1 ADVISABLE OVER Rk WHEN Nc>10^6).`);
l(`Otherwise Rk1 is the same procedure as Rk.`);
    elif i<78 then
l(`neg(z)=nz         Neg(z)=NULL`);
l(`   calculate -z where z=[x,y] or [x,y,n] is a point on E`);
l(`Like Eadd and eadd, Neg and neg accept a "general" point.`);
    elif i<82 then
l(`bas(args)=qR       Bas(args)=NULL   --- args is optional`);
l(`   note: the old names ind,Ind are no longer used.`);
l(`If nargs=0 then BASL:=2000;else BASL:=args; BASL limits the range of the`);
l(`search procedure ouu which is called by bas, to avoid excessively long`);
l(`searches. Clock can also be used.`);
l(`Bas attempts to find Mordell-Weil basis of E(Q) (mod torsion), as itemized`);
l(`below. When successful (qR=5) or possibly successful (qR=3 or 4), bas`);
l(`invokes shaf_tr which inspects all available values of shaf (=order of `);
l(`Shaf.-Tate group) of isogenous curves. When at least one such shaf is`);
l(`available, this gives a value of shaf of the present curve. When the`);
l(`latter is not (approx.) the square of an integer the putative basis is`);
l(`not in fact a basis, and qR is set = 5/2 or 9/4: see the table of qR`);
l(`below. In any case RETURN(qR). `);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`  1. Determine the rank r4, if not already done. Bas may choose to call`);
l(`respectively, rkNC, rk, rk1 or rk2 (resp., to improve rc, if Nc<10^6,`);
l(`if Nc<10^20, if Nc>10^20). RETURN(qR) if unsuccessful. For rc see Menu(rk)\
.`);
l(`  2. If r4=0 then set qR:=5 and wrap up with a call to shaf_tr as `);
l(`described above, and RETURN(qR);`);
l(`  3. Search for independent points. These are stored in the list RR`);
l(`in order of increasing canonical height: ht(RR[1])<ht(RR[2])<...`);
l(`If insufficiently many are found, then RETURN(qR).`);
l(`  4. (A prolog to step 5.) The number of RR[i] that are definitely `);
l(`successive minima (see section 3.5.2 of ECH:= Elliptic Curve Handbook,`);
l(`available by anon. ftp from same site as apecs in the directory pub/ECH1)`);
l(`is denoted RRsm. That is, RR[1],..,RR[RRsm] are succ. min.`);
l(`as determined with the aid lowB = Siksek's lower bound (given by the`);
l(`proc. hdif_Sik()) for h^-h/2 where h^ is canon. height and h is`);
l(`logarithmic naive height. Thus bas must ouu(evalf(exp(2*(h^-lowB))))`);
l(`to ensure that all points with canon. height < a given h^ have been`);
l(`found. In particular, take h^=ht(RR[i]) to test whether RR[i] is the i-th`);
l(`succ. min., except when r4=1 one can take h^=ht(RR[1])/4. `);
l(`Bas needs RRsm>=1. If RRsm=0 then RETURN(qR).`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`  5. Calc. the height pairing matrix B of RR[1],..,RR[r4]. If during this`);
l(`calc. an example of ht(RR[i]+RR[j])<ht(RR[i]), j<i, is found then RR`);
l(`is refined (since RR[i]+RR[j] is a better candidate than RR[i] for the`);
l(`i-th succ. min.) and this step is started afresh. `);
l(`  6. If RRsm=r4 and r4<5 then a theorem of Minkowski applies (sec. 3.5.2`);
l(`ECH) and bas wraps up with qR:=5;shaf_tr();RETURN(qR);`);
l(`  7. Calc. an upper bound, denoted ub_index, of the index of the subgroup`);
l(`generated by the RR[i] and the torsion points in E(Q). The ingredients`);
l(`are: det(B), for i>RRsm the lower bound ht(RR[RRsm]) for the i-th succ.`);
l(`min., and standard constants from the theory of pos. def. quad. forms`);
l(`--- for this I can only refer to sec. 3.5.2 of ECH.`);
l(`  8. If ub_index=1 bas wraps up with qR:=5;shaf_tr();RETURN(qR);`);
l(`Otherwise apply Siksek's sieve (sec. 3.6 ECH). If that eliminates all`);
l(`prime divisors of ub_index then bas wraps up as above.`);
l(`  9. If the sieve is inconclusive then apply half() to linear comb.'s`);
l(`of members of RR. If this discovers an improvement to RR then the`);
l(`process is begun anew at step 5.`);
l(`  10. If BASL allows further searching, bas tries to improve RR and`);
l(`RRsm. If successful then back to step 5. Otherwise RETURN(qR);`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`The (higher) values of qR denote the outcome of bas:`);
l(`  -1 | isogenous curves not determined therefore rk and bas not done`);
l(`   0 | isogenous curves are known from a twist of E (data for E but`);
l(`     |   not for isog. E'=E/~ stored in nisog, etc.)`);
l(`   1 | isog done but bas not yet called`);
l(`   2 | bas done (hence rk hence isog done) but short of indep. points`);
l(` 9/4 | with existing RR, sha has calc. shaf=1/n^2 ==> RR union PP`);
l(`     |   generates subgroup of index n or 2n,..`); 
l(` 5/2 | bas done and have right number of ind. points - but not a basis`);
l(`     |   because yR is not empty (yR = list of points rationally but not`);
l(`     |   integrally dependent on RR)`);
l(`   3 | RR is possib. a basis and reg is possib. the regulator`);
l(`   4 | RR consists of successive minima--prob. a basis (rank>4 ---`);
l(`     | when rank<5, qR can be bumped to 5 by a thm. of Minkowski)`);
l(`   5 | RR "certified" as a basis. But note that if rc<4 this certification`);
l(`     | depends on one or more of the standard conj.'s --- used to obtain \
an`);
l(`     | upper bound for the rank r4 (see Menu(rk) for details about rc).`);
l(`     | In feasible cases rc can be increased by the command Crem.`);    
    elif i<86 then
l(`lin(args)=adet         Lin(args)=NULL`);
l(`    note: the old names gram, Gram are no longer used.`);
l(`args is a sequence or list of points P1,.. on the present curve E.`);
l(`E must be defined over a field K for which tor is programmed:`);
l(`K=Q, Q(T) or Q(sqrt(m_)), i.e., the field label K_ (which is an actual`);
l(`apecs variable --- see Menu(Ell)) has the value K_=1 (or K_=0 and`);
l(`a1,..,a6 are in Q), K_=T or K_=m_. The Pi are required to be defined`);
l(`over K.`);
l(`The RETURN value of lin is adet; this is a global variable, available`);
l(`after either Lin or lin:`);
l(`--- adet=-1 means the proc was aborted either because E is not of the`);
l(`right type or something is wrong with args.`);
l(`--- adet=0 means the points are dependent mod torsion: there exists a`);
l(`linear relation c1*P1+c2*P2+..+ci*Pi=TP, a torsion point, where the ci`);
l(`are integers, not all 0. (When K_=1, c1,c2,.. are available as the apecs`);
l(`global variables num1,num2,.. and TP as TP, which may = O; but these`);
l(`data are not available when K_<>1. In any case the capitalized command`);
l(`Lin displays the relation).`);
l(`--- adet=a positive number means that the Pi are independent mod torsion.`);
l(`In the case K_=1, adet=the canonical height-pairing grammian determinant`);
l(`|<Pi,Pj>|.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`When K_<>1 and lin detects independence mod torsion, adet is given the`);
l(`symbolic value 1. `);
l(`Lin will not fail when K_=1 or when K_<>1 and the Pi are in fact indep.`);
l(`mod torsion. However lin may be uncertain of dependence in some examples `);
l(`when K_<>1:`);
l(`--- adet=infinity means that the result is inconclusive --- the Pi `);
l(`may or may not be dependent mod torsion.`);
l(`    Specifically, when the Pi are dep. then there exists a relation`);
l(`sum_{j in b}P_j=T+2*Q where b is nonempty, T is a torsion point and`);
l(`Q is a lin comb. of (all) the P_i; lin only tests those comb's`);
l(`whose coeff.'s are from {-1,0,1}. Canon. height is not programmed for`);
l(`Q(T) and Q(sqrt(m_))!`);
l(`    For all E over any K a list RR of points that one hopes are indep.`);
l(`mod torsion is maintained by apecs. When K_=1 the independence is`);
l(`guaranteed. In the cases K_=T, K_=m_ and K_=0 with the ai in Q, RR is`);
l(`at least lin-tested with adet>0, possibly infinity. In all other cases`);
l(`a modest attempt is made to prevent dependence: a new candidate point`);
l(`is checked not to be a member of the auxilliary list uR which consists`);
l(`of lin. comb.'s, "within reason", of Pi already in RR.`);
    elif i<88 then
l(`abel(args)=nabel`);l(`Abel(args)=NULL`);
l(`  args is one of: NULL, a rational number, a sequence of five rational`);
l(`Weierstrass coeff's a1,..,a6 or a list [a1,..,a6] of such.`);
l(`Correspondingly j is set = the j-invariant of the present curve,`);
l(`(denoted jay in apecs), args, or the j-invariant of the curve with`);
l(`the given Weier. coeff's.`);
l(`This procedure tells you whether there exists or not (nabel=[] or [[p,\
n]..])`);
l(`an abelian extension of Q over which E acquires everywhere good or`); 
l(`multiplicative reduction; this only depends on j -- the answer is the`);
l(`same for twists. It is also for the same for isogenous curves,`);
l(`even tho' they have different j (unless there's CM).`);
l(`The response is yes (resp. no) iff`);
l(`the answer is yes for all local fields Q_p (resp. no for some`);
l(`local field), and we say correspondingly that j is abelian or non-abel.`);
l(`(at p) -- the analog of the Hasse principle applies here.`);
l(`When non-abel, nabel is a list of pairs [p,n] where p is a prime`);
l(`at which E is non-abel. and the type of failure at p is coded by n:`);
l(`v(x) denotes the p-adic value of x (the value v of the procedure val(x)`);
l(`when p is preset), and j' denotes j-1728.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(` n |       p | non-abel. behavior at p`);
l(`___|_________|____________________________________________________`);
l(`-2 |       2 | j=1728`); 
l(`-1 |       3 | j=0`);
l(` 1 | 3 mod 4 | v(j)=0, v(j')=1 mod 2`);
l(` 2 | 5 mod 6 | v(j)>0, v(j)<>0 mod 3`);
l(` 3 |       3 | v(j)>0, v(j')=1 mod 2`);
l(` 4 |       3 | v(j)=2, j/9<>1 mod 3`);
l(` 5 |       3 | v(j)=3, j/27<>4,10,13 or 22 mod 27`);
l(` 6 |       2 | v(j)>0, v(j)<>0 mod 3`);
l(` 7 |       2 | 0<v(j)<12,v(j)=0 mod 3 and either v(j)<>6 or`);
l(`   |         | v(j)=6 and j/64<>1 mod 8`);
    elif i<90 then
l(`Roha()=NULL,      roha()=roh=1 or -1`);
l(`  roh = the sign of the functional equation calculated as a product of lo\
cal`);
l(`root numbers.`);
l(`The variable SFE, when <>0, denotes an earlier determination of this`);
l(`sign either analytically by Sfe() or by SFE=(-1)^rank. Of course all of`);
l(`this is conjectural. When both roh and SFE are nonzero they are compared`);
l(`-- they should be equal -- alarm bells if not. At the end roha() sets`);
l(`SFE:=roh.`);
l(`   The value of SFE is transferred to isogenous curves when these have`);
l(`been determined by apecs.`);
l(`   The procedures Roha and roha, which determine the root number in all`);
l(`cases, supersede the earlier procedures Rohr and rohr which often failed`);
l(`when there was additive reduction at 2 or 3.`);
    elif i<92 then
l(`Absc(x) = NULL        absc(x) = [x,y]`);
l(`  Find a point [x,y] on E with given abscissa x; x needn't be rational.`);
l(`Of course this simply requires solving a quadratic equation for y.`);
l(`Only one of the (usually) two possibilities is given.`);
l(`Examples:  with cur=A11, Absc(1) ===> [1,0], Absc(2) ===>`);
l(`[2,-1/2+1/2 17^(1/2), and t:='t':gpt:=absc(t);  makes gpt a generic point.`);
    elif i<94 then
l(`Negp(pz)=NULL         negp(pz)=nz`);
l(`  Calculate the negative nz of a point pz on E mod p. The prime p`);
l(`must be preset.`);
    elif i<96 then
l(`Search(args)=NULL`);
l(`search(args)=searchl             See also Seek and Seek1.`);
l(`  Let E be defined over Q either by Ein (so K_=1) or Ell (so K_<>1).`);
l(`  Search for rational points [x,y] on E with max(|numer(x)|,denom(x))`);
l(`between args[1] and args[2] (or between 1 and args if nargs=1) and`);
l(`store all points found in list searchl. No x-axis adjustment is made as`);
l(`in Seek and, when K_=1, no regard is paid to the list RR of independent`);
l(`points or the list uR of misc. points --- to save any possibly independent`);
l(`points use map(App,searchl). Note that only one of a pair +-P is included,`);
l(`and another call to Search will start from scratch.`);
    elif i<98 then
l(`Seek1(args)=IP`);
l(`seek1(args)=nops(IP)              See also Seek and Search`);
l(`  Let E be defined over Q either by Ein (so K_=1) or Ell (so K_<>1).`);
l(`  Look for integral points [x,y] on E with |x| (adjusted by "f" as in`);
l(`Seek) between args[1] and args[2] (or up to args if nargs=1) and`);
l(`store points found in list IP.`);
l(`When K_=1:  also include in IP any integral points`);
l(`  accumulated in the "misc." list uR -- this may include some with |x|`);
l(`  outside the range specified by args. Also process these points by "ap1_"`);
l(`  as in Seek, but don't increase the "Seek limit" parameter sB since`);
l(`  possible points with denom(x)=4,9,16... haven't been checked.`);
l(`When the Weier. coeff.'s ai are not all integral (hence K_<>1), seek1`);
l(`clears denom.'s by transforming (temporarily) ai|-->ai*uV^i where the`);
l(`pos. integer uV is minimal. This amounts to substituting in the Weier.`);
l(`equation x=X/uV^2, y=Y/uV^3. Then seek1 puts in the list searchl the`);
l(`[x,y] found such that [X,Y] is integral, whereas the list IP contains only`);
l(`[x,y] that are actually integral (together with their negatives when`);
l(`integral). Thus one would use`);
l(`         ein(cur);for z in searchl do trcw(z);od`);
l(`when looking for indep. points on the Ein curve.`);
    elif i<100 then
l(`Raf()=NULL`);
l(`raf()=raF`);
l(`   Calculate list raF of x-coordinates of real (resp. all) points of`);
l(`order 2 when E is a catalog curve, i.e. the variable cur has a value`);
l(` such as A11 (resp. not a cat. curve and cur='cur').`);
    elif i<102 then
l(`We() or We(args)=NULL`);
l(`we() or we(args)=wec`);
l(`   args specifies a catalog curve; no args means take present curve.`);
l(`args (if present) may have any of the forms C33 (a catalog name), 65`);
l(`(a cat. number),[1,1,0,44,55] or 1,1,0,44,55 (a list or seq.of coef.'s).`);
l(`The 5 Weierstrass coef.'s of the specified curve are put in the list`);
l(`wec. Thus in the case nargs=0, wec=[a1,a2,a3,a4,a6].`);
l(`We displays the Weierstrass equation; we displays wec.`);
    elif i<104 then
l(`Div(args)=NULL,         div(args)=al.args`);
l(`   Calculate the polynomial al.args e.g. if args=6 then calculate al6.`);
l(`These polynomials are essentially the division poly.'s psi_n`);
l(`     al.n=psi_n (n odd),   al.n=psi_n/(2y+a1x+a3) (n even)`);
l(`They are calculated recursively (see op(div)); the variable aln is the`);
l(`highest n for which al.n has been calculated. However not all lower al.i`);
l(`will have been calculated in general --- only up to i=trunc(n/2)+2`);
l(`since that's all that's needed to calculate al.n by the recurrence.`);
l(`   apecs begins with aln:=4 and`);
l(`      al1:=1:    al2:=1:`);
l(`      al3:=3*x^4+b2*x^3+3*b4*x^2+3*b6*x+b8:`);
l(`al4:=2*x^6+b2*x^5+5*b4*x^4+10*b6*x^3+10*b8*x^2+(b2*b8-b4*b6)*x+b4*b8-b6^2:`);
l(`also al0:=(4*x^3+b2*x^2+2*b4*x+b6)^2: (a constant used in the recursions).`);
    elif i<106 then
l(`Tri(args)=NULL          tri(args)=trz`);
l(`  where args is either a pair z,oc or just oc where z is a point on E`);
l(`and oc is a curve isogenous to E: calculate the image point trz on oc`);
l(`E.g. Ein(D91);Tri([15,24],C91); Equivalently, since C91 has catalog number`);
l(`ncur=271, Tri([15,24],271);.`);
l(`   This has now been implemented for all cyclic isogenies of degree<=27;`);
l(`for composite degrees the "cheapest" path through prime degrees is taken.`);
l(`Calculate the transformation equations of the isogeny and store them in`);
l(`visog (see Menu(isog)) if they are not already there; then use these`);
l(`equations to calculate the image trz of the point z. This last`);
l(`calculation is omitted if z is absent from args, and then trz='trz'.`);
l(`When trz is calculated it is adjoined to RR when that is appropriate,`);
l(`as described in Menu(App).`);
    elif i<108 then
l(`Velu(args)=NULL           velu(args)=ncur`);
l(`   Determine the isogenous curve E'=E/G which becomes the present curve`);
l(`   where G is the subgroup determined by args.`);
l(`We have begun to make velu robust so that E can be defined over`);
l(`"any" field K of char. 0. args is either a sequence,list or set`);
l(`of points, in which case G is the subgroup of E(K) generated by these`);
l(`points, or args is a polynomial in X defined over K whose roots(possibly`);
l(`algebraic over K) are distinct and are the X-coordinates of the points`);
l(`in G. Thus if args=(X-x1)*..(X-xn) then |G| is 2n or 2n+1 and`);
l(`G={O,+-[x1,y1],..}`);
l(`    It is convenient to describe separately two cases:`);
l(`    CASE: E is a catalog curve (old or new). Then necessarily K=Q, and`);
l(`the value of velu(args) is the catalog number of E'. Also calculate`);
l(`the transf. equations of the isogeny and store in the variable visog`);
l(`(see Menu(isog)) attached to E.`);
l(`Examples: The torsion subgroup/Q of E=A80 consists of O and PP=[0,0,2].`);
l(`Velu(z) where z is any one of [0,0,2], [0,0], 0,0, or X produces E'=B80.`);
l(`But to get the 3-isogenous curve C80 we can only take z=X+1. (The kernel`);
l(`is {O,[-1,I],[-1,-I]}, as we see by the command factor(al3) ---`);
l(`as explained by Nota(), al.n is essentially the n-division polynomial.)`);
l(`   Of course in this case, Isog() determines all curves Q-isogenous to E.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   CASE: E is not a catalog curve. This case is characterized by`);
l(`cur='cur' is unevaluated. K=Q is not precluded --- E/Q can be introduced`);
l(`by Ell. |G| is bounded by Nsubg, whose default value is 16, and if`);
l(`exceeded the procedure is aborted. This variable can be assigned another`);
l(`value at any time. The image under the isogeny of the general point [X,Y]`);
l(`of E is [x7,y7] in E'(K), where x7, y7 are rational functions in X, Y.`);
l(`Thus to obtain the image of a point [a,b] one puts X:=a;Y:=b; so the`);
l(`image point is [x7,y7]; DON'T FORGET to finish up with X:='X';Y:='Y';`);
l(`x7,y7 are not saved in anything like visog as in the catalog curve case,`);
l(`so its best to store their values in other variables if they will be`);
l(`needed later.`);
    elif i<110 then
l(`Exam() or Exam(args) =NULL;   exam() or exam(args)=NULL`);
l(`   Exam is a repository for examples and formulas, and also an input`);
l(`procedure for curves with specified torsion.`);
l(`Exam(); displays an index of the various Exam topics/functions.`);
l(`Exam(i); displays the data for topic #i. exam(i); is a shortened form.`);
l(`Exam(n,t); (resp. Exam(2n,1,t)) inputs a curve with torsion group`);
l(`Cn (resp. C2xCn) determined by the parameter t -- with slightly`);
l(`different args for C3 and C2xC2:`);
l(`   For C3 use Exam(3,a1,a3);`);
l(`   For C2xC2 use Exam(4,a2,a);`);      
l(`See Exam(1) for further details and examples.`);
    elif i<114 then
l(`Hdif_Sik()=NULL;     hdif_Sik()=lowB`);
l(`Hdif_Sil()=NULL;     hdif_Sil()=[lowB_Sil,uppB]`);
l(`Let P be any rational point on E, let h denote its logarithmic height`);
l(`i.e. if P=[x,y] and x=a/b with gcd(a,b)=1, then h=log(max(|a|,|b|)),`);
l(`and h^ its canonical (Weil-Neron-Tate) height (as can be calculated by`);
l(`the procedure ht(x)). Then for all P,`);
l(`        max{lowB,lowB_Sil} < h^ - (1/2)h < uppB`);
l(`It seems lowB is usually better (bigger) and so the procedure`);
l(`bas calls hdif_Sik().`);
l(`Ref's: J. Silverman, "Difference between Weil..", Math. Comp. 1990, 723-\
743;`);
l(`S. Siksek, "Infinite descent on ell. curves", Rocky Mountain J. Math.,`);
l(`25(1995), 1501--1538.`);
    elif i<116 then
l(`RkNC(args)=NULL;    rkNC(args)=NULL;   -- usually args is absent`);
l(`Call rk(args) to calculate the rank with rflag=3 so that No Conjectures`);
l(`are used -- no use of Mest(), FnL() or (-1)^r4=SFE.`);      
l(`This can be used, e.g., when 0<=rc<4 (which means that r4 is only a`);
l(`conjectural value of the rank -- see Menu(bas) for more details about rc)`);
l(`to attempt to increase rc; of course it can also be used when rc=-1.`);
    elif i<118 then
l(`Expr(z)=NULL       expr(z)=[num1,..,num.MM],TP  or  z  or NULL`);
l(`Express z=num1*P1+.. as a linear combination of the points P1,.. in RR`);
l(`+ TP, a torsion point -- when possible. When z itself is a torsion point`);
l(`expr returns z; when only a multiple of z, denoted den*z, is expressible`);
l(`thus, expr returns [num1/den,..],TP and then calls ap1_(z) which attempts`);
l(`to improve RR, i.e. smaller height pair. det. -- so the fractions`);
l(`num1/den,.. will not be relevent to the new RR; when z is indep. of RR`);
l(`mod torsion expr returns NULL, and in this case also calls ap1_(z) which`);
l(`results in a new RR of length one greater.`);
    elif i<120 then
l(`Store()=NULL       store()=stORe`);
l(`This command toggles the storage mode of apecs between "store all curves`);
l(`in the stack and also, when K_=1 (see Menu(ell) for K_), in the catalog"`);
l(`(then the variable stORe=1 --- apecs begins this way),`);
l(`and "store only the present curve and possibly some isogenous`);
l(`curves" (then stORe=0, and these will be overwritten when a new curve`);
l(`is introduced). This is useful when large families of curves are`);
l(`being processed so that maple does not become bogged down. One can`);
l(`toggle Store() to keep only the interesting curves in the catalog.`);
l(`Of course one must remember to use Zpecs(); or Ypecs(); to save the`);
l(`enlarged catalog if these curves are wanted for another session.`);
    elif i<122 then
l(`Trans(r,s,t,u)=NULL    trans(r,s,t,u)=NULL    or`);
l(`Trans(r,s,t,u,k)=NULL    trans(r,s,t,u,k)=NULL`);
l(`5-th arg is optional `);
l(`   Transform the present Weierstrass equation`);
l(`        y^2+a1*xy+a3*y=x^3+a2*x^2+a4*x+a6`);
l(`by substituting`);
l(`      x=u^2*x'+r,  y=u^3*y'+s*u^2*x'+t`);
l(`and then invoking  Ell(or ell)(a1',..,a6',K_)  (see Menu(ell) for K_)`);
l(`where K_ has either its present value or, when there is a 5-th arg, is set`);
l(` := k. (Exceptions: 1. When present K_=1 then K_ is set = 0 since the`);
l(`transformed E is not Weier. reduced. 2. If the transformed curve is not`);
l(`defined over the field with label K_ then K_ reverts to 0.)`);
l(`The coefficients a1',.. of the resulting Weierstrass equation are`);
l(`     u*a1'= a1+2*s`);
l(`   u^2*a2'= a2-s*a1+3*r-s^2`);      
l(`   u^3*a3'= a3+r*a1+2*t`);
l(`   u^4*a4'= a4-s*a3+2*r*a2-(t+r*s)*a1+3*r^2-2*s*t`);
l(`   u^6*a6'= a6+r*a4+r^2*a2+r^3-t*a3-t^2-r*t*a1`);
l(`Both the original and the transformed E are in the stack and Trwc and`);
l(`Trcw (see Menu(Trcw)) are available for both.`);
l(`Note also that Trans and trans automatically transfer points from RR`);
l(`of the original E to RR of the transformed E.`);
    elif i<124 then
l(`Cubic_fit(args)=NULL,      cubic_fit(args)=f_`);
l(`***** These commands may result in an ERROR message ---`);
l(`      then simply repeat the command verbatim, or precede the`);
l(`      the cubic_fit command with prep_(); see below for explan.*****`);
l(`   where args consists of two parts, both optional:`);
l(`part 1 is a sequence of points [u1,v1],...,[un,vn] (n >= 0)`);
l(`in the affine plane, and part 2 (if present) is either a single`);
l(`equation, or a set or a list of equations involving the variables`);
l(`s_1,..,s_9 of the form s_i=... . These procedures calculate a cubic`);
l(`f_:=s_1*u^3+s_2*u^2*v+_s_3*u*v^2+s_4v^3+s_5*u^2+s_6*u*v+s_7*v^2+s_8*u+s_\
9*v`);
l(`passing through the points [u,v]=[0,0] and [ui,vi], i=1..n,`);
l(`and whose coefficients satisfy the equations in part 2 of args.`);
l(`EXAMPLE: Cubic_fit([0,1],[1,1],[1,2],[2,-1],[3,-1],{s_1=1,s_7=s_8,s_9=\
1/3})`);
l(`The result may be "no solution" -- then f_=NULL -- or some of the s_i`);
l(`may be left as undetermined (parameters). One can leave them as param.'s`);
l(`or give them values by commands (e.g. s_5:=0; s_6:=my_s6;..)`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   Now one can use Third_pt (see Menu(third_pt)). If all the s_i are`);
l(`rational one can get the birationally equivalent min. Weierstrass equation`);
l(`by invoking Gcub(s_||(1..10)); (or Gcub(s_.(1..10)) in MapleV and`);
l(`earlier). s_10 has automatically been set = 0.`);
l(`Remember that the variable bic contains the details of the birat.`);
l(`correspondence --- see Menu(gcub), and the commands Trwc and Trcw can be`);
l(`used to translate points between Cubic and Weier.`);
l(`***** When cubic_fit is called more than once then (probably) some of`);
l(`the s_i have been assigned values which may collide with equations`);
l(`in "part 2" of args --- and I can't program around this because the`);
l(`args are evaluated before anything in the procedure is carried out.`);
l(`Repeating the command clears out the confusion because cubic_fit`);
l(`initially sets all s_i to unevaluated.`);
l(`    Alternatively one can precede the cubic_fit command with prep_();`);
l(`which is an internal apecs procedure that sets s_1:='s_1': etc.`);
l(` --- e.g. in a loop in your own program calling cubic_fit you should`);
l(`have   ... prep_();cubic_fit(...); ...  .`);
    elif i<126 then
l(`Third_pt(z1,z2)=NULL,     third_pt(z1,z2)=[u3,v3]`);
l(`   Calculate the third point z3 = [u3,v3] of intersection of the line`);
l(`joining the two given zi = [ui,vi] (z1 = z2 is allowed) with the`);
l(`cubic f_=s_1*u^3+...+s_9v (s_10=0) which is normally preset by the`);
l(`command Cubic_fit (see Menu(cubic_fit);). These commands encompass`);
l(`only affine points, and so one cannot have z1=[], for instance.`);
l(`On the other hand the result may be "at infinity" --- then z3 = [].`);
    elif i<128 then
l(`Weier_fit(args)=NULL      weier_fit(args)=f_`);
l(`***** These commands may result in an ERROR message ---`);
l(`      then simply repeat the command verbatim. Alternatively,`);
l(`      precede the Weier_fit command with the apecs command wprep_();`);
l(`      The situation is similar to that of Cubic_fit.`);
l(`      See Menu(cubic_fit) for explan.*****`);
l(`   where args consists of two parts, both optional:`);
l(`part 1 is a sequence of points [u1,v1],...,[un,vn] (n >= 0)`);
l(`in the affine plane, and part 2 (if present) is either a single`);
l(`equation, or a set or a list of equations involving the variables`);
l(`a_1,a_2,a_3,a_4,a_6 of the form a_i=... . These procedures calculate`);
l(`        f_:=x_^3+a_2*x_^2+a_4*x_+a_6-y_^2-a_1*x_*y_-a_3*y_`);
l(`passing through the points [ui,vi], i=1..n, and whose coefficients`);
l(`satisfy the equations in part 2 of args.`);
l(`There may be no solution, and then f_=NULL is returned, or some of the`);
l(`a_i may be left as parameters. When the parameters (if any) are given`);
l(`rational values, one can, e.g., call Ein(a_||(1..5)); (or a.(1..5) in`);
l(`MapleV and earlier). a_5 is auto. set = a_6 by weier_fit.`);
l(`EXAMPLE: Here is how the examples of T (torsion subgroup) = C2 and`);
l(`r (rank of E(Q)) = 4 and 5 in the table displayed by Exam(4) were found:`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`The Weierstrass equation`);
print(Y^2+a1XY=X^3+a2X^2+a4X);
l(`has the point [0,0] of order 2. We use weier_fit to make this curve go`);
l(`thro' 3 integral points, which often gives r>=3, and look for cases r>3:`);
l(`   ppts:=NULL:#ppts will be set of all integral points [x,y] with`);
l(`              #-2<=x,y<=2, [0,0] excluded`);
l(`   for x from -2 to 2 do for y from -2 to 2 do ppts:=ppts,[x,y]:od:od:`);
l(`   ppts:={ppts} minus {[0,0]}:`);
l(`   with(combinat):pts3:=choose(ppts,3):#pts3 is set of triples of`);
l(`              #distinct points from ppts`);
l(`   r4l:=[]:#r4l will be list of cases with (probably) r>3`);
l(`   for pt3 in pts3 do`);
l(`     wprep_():weier_fit(op(pt3),{a_3=0,a_6=0}):`);
l(`     if f_<>NULL and type(f_,polynom(rational)) then ein(a_||(1..5)):`);
l(`       if DD<>0 then rkNC():`);
l(`         if r4>3 then r4l:=[op(r4l),pt3]:fi:`);
l(`     fi:fi:`);
l(`   od:`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Sometimes different triples pt3 give the same curve E. For r=4 we`);
l(`found 11 different E -- Ein(625) is the one with smallest conductor`);
l(`Nc -- and one E with r=5, namely Ein(626).`);
    elif i<130 then
l(`Tor()=NULL      tor()=NN`);
l(`Tor(z)=NULL     tor(z)=NN`);
l(`Determine the torsion subgroup of E(K) where E is an elliptic curve`);
l(`defined over the field K in the following cases:`);
l(`--- K=Q, any field label K_ (see Menu(ell) for K_)`);
l(`--- K=Q(sqrt(m_)) with K_=m_ (see Menu(qfin))`);
l(`--- K=Q(T) with K_=T (the variable must be T)`);
l(`Normally E/Q are introduced by Ein (or Ein0), and in the other cases by`);
l(`Ell with the extra parameter K_=m or T (again see Menu(Ell) for details).`);
l(`Other venues for the intro. of E are Genj, Quar, Gcub, Velu,.. .`);
l(`   The seldom used optional param. z -- available only in the case K=Q --`);
l(`is z = a torsion point [x,y] that is somehow known beforehand.`);
l(`(This feature is exploited by the commands exam(n,t), etc.)`);
l(`Notation:   NN=order of torsion subgroup`);
l(`            PP=sequence of non-O torsion points in the format [x,y,n]`);
l(`where n=the order of the point. Thus PP=NULL when NN=1.`);
l(`            ouP=[...,[x,y],...] list of members of PP with n dropped`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`NN=0 signals that tor is not programmed for the present curve, e.g. when`);
l(`K_=0 and some ai is irrational. NN=-1 means that tor has not yet been`);
l(`called. Even when NN=0, PP and ouP are maintained: tor finds the points`);
l(`of order 2 defined over the field generated over Q by the Weier. coeff.'s`);
l(`a1,..a6. (but NN remains 0); and PP, ouP are augmented`);
l(`by any point of finite order and its multiples that happens to be`);
l(`discovered (perhaps by a call App(x,y)).`);
l(`   The general approach is to find the roots in K first of prac for`);
l(`points of order 2, then of the division poly.'s ali for appropriate`);
l(`i=3,4,.. guided by an upper bound for NN usually obtained by reduction`);
l(`of E mod "primes", i.e., principal prime ideals. Note that these`);
l(`reductions usually involve internal calls to ein or ell which result in`);
l(`unexpected entries in the stack; see Menu(Go). tor calls torq when K_=m_`);
l(`(see Menu(Qfin)) or torT when K_=T. In the latter case, the higher`);
l(`division poly.'s easily get out of control, and so torT is based on the`);
l(`Nagell-Lutz approach.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    In the case K_=1, various techniques are potentially employed by tor:`);
l(`clues from the Tate alg. or from isog. curves; reduction mod good and`);
l(`bad p; etc. Nagell-Lutz is there as a last resort, but is seldom used`);
l(`because with even moderately sized coeff.'s, there can be an impractically`);
l(`long list of candidate divisors.`);
l(`EXAMPLE 1:  Ein(F112);Tor();# the catalog curve F112 has torsion group/Q`);
l(`                            # C2 = cylic order 2`);
l(`            Ell(cur,2);Tor();# F112 has torsion group/Q(sqrt(2)) C2xC2`);
l(`EXAMPLE 2:  Ell(0,sqrt(8),0,3/2,0);Tor(); ===> NN=0 and`);
l(`            PP:=[-3sqrt(2)/2,0,2],[-sqrt(2)/2,0,2],[0,0,2]`);
l(`            ("possibly incomplete").`);
     elif i<132 then
l(`Qfin(m)=NULL      qfin(m)=m_`);
l(`   Point apecs to the quadratic field Q(sqrt(m)); m must be a squarefree`);
l(`integer <>1. qfin (which stands for quadratic field initialization)`);
l(`causes apecs to:`);
l(`set m_:=m (also K_:=m, cf. Menu(ell);) and initialize apecs to treat ell\
iptic curves defined ov\
er Q(sqrt(m_)). (Many of the names used in this package end in an under\
line _ to make them distinctive). An integral basis is 1,w_ where w_= s\
qrt(m_) or (-1+sqrt(m_))/2.  w_ is a root of pol_.  ram_ is the set of ramifi\
ed primes. h_ is the class number "indicator": h_=1 means class no.=1, h_=2 \
means class no. >1, h_=0 means class no. unknown. Some tables of soln.'s to c\
ertain congruences (e.g.x^4==a+b*w_ mod 8) are set up.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   In practice, often qfin is only invoked implicitly by the command Ell`);
l(`That is, when Ell is given appropriate args, it automatically calls qfin.`);
l(`See Menu(ell);.`);
l(`   Other procedures in this package are (all in the apecs file QF_):`);
l(`--- Fundunit(): when m_>0 calculates the fundamental unit, i.e., the`);
l(`smallest unit > 1, in the notation unit_+vnit_*w_ with approx. real value`);
l(`enit_ and with norm nnit_ = +-1.`);
l(`(fundunit is called automatically by minf when m_>0.)`);
l(`--- nm_(z) calculates the norm of z`);
l(`--- Fact(z) factorizes the alg. integer z; see Menu(fact)`);
l(`--- Minf(): After a curve E defined over Z[w_] has been introduced by`); 
l(`Ell, this proc. transforms E to global minimal Weierstrass form ---`);
l(`programmed ONLY FOR h_ = 1. See Menu(minf).)`);
l(`--- Torq() calculates the torsion subgroup of E(Q(sqrt(m_)))`);
l(`    (any h_); but usually one calls Tor() (which calls torq) --- see`);
l(`    Menu(torq); or, less ambitiously,`);
l(`--- Ub_tor(args) calculates an upper bound for the order of the torsion`);
l(`    subgroup by reducing E mod p for the first n primes p which do not`);
l(`    divide nm_(DD) and which are either ramified or split, where n=args`);
l(`    (default n=4 when args is absent). ub_tor is called by torq, and`);
l(`    and Torq/torq can have an args which is passed to ub_tor.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Here is a sample session:`);
l(`        Ell(0,2*w_,0,3-7*w_,0,5);Minf();Tor();`);
l(`The 6-th arg of Ell, 5, causes ell to call qfin(5); see Menu(ell);.`);
l(`Ell allows the args a1,..,a6 to be any members of Q(w_,sqrt(m_)). In`);
l(`the above example, since sqrt(5)=2*w_+1, 2*w_ could be replaced by`);
l(`-w_+3*(5/sqrt(5)-1)/2; for more detail, again see Menu(ell).`);
    elif i<134 then
l(`Minf()=NULL        minf()=NULL`);
l(`The prerequisites to using these commands are: apecs is pointing to a`);
l(`quadratic field Q(sqrt(m_) with h_ =1, i.e., class number = 1, and the`);
l(`present curve is defined over the ring of integers Z[w_] of that field.`);
l(`For notation see Menu(qfin). Usually these prerequisites are met by the`);
l(`Ell command with appropriate args, as explained in Menu(Ell). For example`);
l(`Ell(0,-2,w_,(5-2*sqrt(17))^7,0,17);. Then minf finds a global minimal`);
l(`form E' of E over Z[w_] (existence guaranteed when the class number h_=1)`);
l(`and calls ell(new_,m_) where new_ is the list of Weierstrass coefficients`);
l(`of E' (old_ is the list for E), and calculates the transformation`);
l(`tau_=[r_,s_,t_,u_] from E to E.'`); 
l(`For the theory see Chapter 5, section 5.6.2 of Elliptic Curve Handbook.`);
    elif i<136 then  
l(`Fact(args)=NULL      fact(args)=facs_`);
l(`  It is necessary that K_= some m_ as explained in Menu(ell); see also`);
l(`Menu(Qfin); args must be an algebraic integer in Z[w_], and so (after`);
l(`simplification) has the form a+b*w_ where a,b are integers.`);
l(`  Find the factorization product(p_^{v_}) in Z[w_] of args and put`);
l(`results in the list facs_=[..,[p_,v_],..]. p_ denotes a prime ideal`);
l(`and is actually an alg. integer when the ideal is principal; or may`);
l(`be literal, when [p_,v_] has one of the forms [P.p,v_] (when the`);
l(`rational prime p ramifies as pZ[w_]=(P.p)^2, P.p non-principal), or`);
l(`[P.p,N] or [Q.p,v_-N] where v_ is an actual integer and N is an`);
l(`undetermined integer in the case when p splits as (P.p)(Q.p) into two`);
l(`non-princ. prime ideals.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`EXAMPLES: Qfin(5);Fact(20-30*w_); ==>`);
print(`The factorization of .. up to a unit factor is`);
print([[2,1],[1+2*w_,2]]);
l(`meaning that 20-30*w_=u*2*(1+2*w_)^2 where u is a unit; in fact,`);
l(`(1+2*w_)^2=5 and u=2-3*w_. Since sqrt(5)=2*w_+1, Fact(35-15*sqrt(5))`);
l(`would give the same result. In general, args can be any rational`);
l(`function in w_ and sqrt(m_), as long as it simplifies to the form`);
l(`a+b*w_ with a,b in Z.`);
l(`    On the other hand, Qfin(-5);Fact(20-30*w_); prints out`);
print([[P_2,2],[w_,2],[P_7,N],[Q_7,2-N]]);
l(`meaning that the factorization of the principal ideal (20-30*w_)Z[w_]`);
l(`is P_2^2*(w_)^2*P_7^N*Q_7^(2-N) where P_2 is the unique ramified ideal`);
l(`of norm 2 and P_7, Q_7 are the two ideals of norm 7; N is an undeter-`);
l(`mined integer (one of 0,1,2), and of course depends on how one might`);
l(`explicitly identify P_7 and Q_7.`);
    elif i<138 then
l(`Crem(args)=NULL      crem(args)=r1`);
l(`  with either no args, or args=dil, or args=A or args=A,dil or args=dil,A`);
l(`  where A is the actual letter A. dil has the default value 30.`);
l(`For E defined over Q, the steps performed by Crem are as follows.`);
l(`   1. If E has a rational point P of order 2 then call RkNC(dil) which`);
l(`carries out 2-descent via the 2-isogeny E-->E/<P> (see Menu(RkNC)) ---`);
l(`unless args includes A and then overide this and go to step 2.`);
l(`   2. Apply the algorithm of Birch & Swinnerton-Dyer`);
l(`[J. reine angew. Math. 212(1963)] as refined by Cremona [Algorithms for`);
l(`modular elliptic curves], especially his recently discovered test for`);
l(`the isomorphism of quartics to appear in the new edition of his book,`);
l(`to enumerate the iso. classes of associated quartics V^2=a_*X^4+..+e_`);
l(`hence get an upper bound r1 for the rank.`);
l(`   3. Attempt to find which of these quartics is an elliptic curve`);
l(`(necess. iso./Q to E) i.e. either a_ is a square or there is a rational`);
l(`point. Search these quartics for rational points with X=n/d up to`);
l(`max(|n|,d)<=dil. Sometimes the rank is determined before this search is`);
l(`completed and then the procedure is terminated --- unless the letter A is`);
l(`one of the args and then the search is carried through. The latter can be`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`useful to discover new independent points: the first point found on a`);
l(`quartic is used by the "subroutine" quar to set up an iso. between the`);
l(`quartic and E. Then points found subsequently are mapped to E and`);
l(`processed appropriately. Cf. the apecs command App.`);
l(`   4. The elliptic quartics are stored in eq_, while the undecided`);
l(`quartics are stored in q_. These and other relevant data are stored in`);
l(`the "stack" (but not in cat) so that one can call Crem repeatedly`);
l(`(even after returning to E after considering other curves, e.g. by the`);
l(`command Go): a subsequent call to Crem(dil) will resume the search for`);
l(`rational points (on the quartics in both eq_ and q_) starting at the place`);
l(`left off by the previous dil, without redoing Birch,Swinnerton-Dyer,Cremona.`);
l(`It is not necess. to include A in args -- A is deemed to be present. Note`);
l(`that the stack is preserved only in the present apecs session, so that in a`);
l(`subsequent session Crem will start from scratch.`);
    elif i<140 then
l(`Om2()=NULL           om2()=omegA[1]`);
l(`This command no longer exists --- all the work is done by Om.`);   
    elif i<142 then
l(`Subg(args)=op(GG)       subg(args)=diso`);
l(`    Calc. the subgroup of E(K) generated by args where args is a sequence,`);
l(`list or set of one or more points and where E is defined over`);
l(`the field K of char. 0, e.g. K=Q, Q(T) or an alg. number field. GG is the`);
l(`set of nonzero elements of the subgroup and diso:=nops(GG)+1 is the`);
l(`subgroup order. If during the calc. the set GG obtained so far has`);
l(`cardinality >= the apecs variable Nsubg then the procedure is aborted,`);
l(`though GG is still available`);
l(`Nsubg has the default value 16 (since when K=Q, diso>16 indicates that GG`);
l(`is infinite) and can be assigned another value at any time. The notation`);
l(`used for a point P=[x,y] in GG is [x,y,n] where n is the order of P, or`);
l(`n=0 when the order is infinite or at least > Nsubg.`);
    elif i<144 then
l(`Torq(args)=NULL         torq(args)=NN`);
l(`    Calculate the torsion subgroup of E defined over the quadratic field`);
l(`Q(sqrt(m_)) by first calling ub_tor(args) (args may be absent --- see`);
l(`Menu(ub_tor)), and then finding roots of appropriate division poly.'s.`);
l(`For more details about quad. fields, see Menu(qfin); and for more about`);
l(`the methods for calculating torsion, see Menu(tor);.`);
l(`    Tor automatically calls torq when apecs is pointing to an`);
l(`E/Q(sqrt(m_)), and in that case one may as well simply call Tor();`);
l(`instead of Torq(); the only purpose of keeping torq available as a user`);
l(`command is to allow the possibility of passing args to ub_tor.`);
    elif i<146 then
l(`Ub_tor(args)=NULL       ub_tor(args)=ubNN    args(= pos. integer) optional`);
l(`    Set hex:=args (default hex:=4 if args is absent).`);
l(`    Find upper bound ubNN for torsion of E defined over quadratic field`);
l(`Q(sqrt(m_)) by reducing E mod p for hex successive appropriate primes p.`);
l(`See Menu(qfin) for notation and more details. For the possible kernels`);
l(`of reduction mod p, see Corollary 2.10.5 in Elliptic Curve Handbook.`);   
    elif i<148 then
l(`Go()=NULL             go()=NULL`);
l(`Go(args)=NULL         go(args)=NULL`);
l(`With no args, Go displays the current stack, each line of the display`);
l(`consisting of the curve's stack number (called stac in apecs) and either`);
l(`the curve's name if K_=1, which is stored in ft1[stac], or the 5`);
l(`Weierstrass coeff.'s if K_<>1, which are stored in ft2[stac], and finally`);
l(`the field label K_. (See Menu(ell) for details about K_.)`);
l(`Data pertinent to the curve is stored in ft[stac] to be retrieved if`);
l(`apecs returns to that curve, but is not displayed by Go. (See Menu(Dat);`);
l(`to display the curve's data.)`);
l(`    Go(n) returns apecs to the curve with stac=n, along with previously`);
l(`calculated data that was saved in ft[stac].`);
    elif i<150 then
l(`Fundunit()=NULL       fundunit()=enit_`);
l(`  It is necessary that K_ = some positive m_, as explained in Menu(ell).`);
l(`See also Menu(qfin); for more on the notation.`);
l(`   Calculate the fundamental unit, unit_+vnit_*w_, of Q(sqrt(m_), and its`);
l(`absolute value (i.e., a real approx.) enit_ and its norm nnit_= +-1.`);
l(`"Fundamental" means the smallest unit with enit_>1.`); 
    elif i<152 then
l(`Cld()=NULL           cld()=uV`);
l(`For an E/Q, clear denominators of Weier. coeff.'s by ai|-->ai*uV^i`);
l(`where pos. integer uV is minimal. The resulting curve/Z becomes the`);
l(`present curve with field label K_=0. To have this curve treated as a`);
l(`catalog curve, use the command Ein(cur); it will then be put in min.`);
l(`Weier. form and will have field label K_=1.`);
    elif i<154 then
l(`Isom(args)=NULL         isom(args)=isom_list or false`);
l(`args=n1 or n1,n2   --- one or two stack numbers (see below)`);
l(`  Given one curve (or two curves) over the field K where K is either`);
l(`Q or Q(sqrt(m)) or the rational function field Q(T), i.e., the field label`);
l(`of the curve (or of both curves) K_=1, m or T (see Menu(ell)), find all`);
l(`the automorphisms (or the isomorhisms) defined over K. In the latter case`);
l(`RETURN(false) if there are none; else list all the auto.'s (or isom.'s)`); 
l(`in the variable isom_list in the standard notation [r,s,t,u]`);
l(`(so u*a1'=a1+2s, etc.). Cf. Menu(Trans). Isom allows a mix of K_=1 for`);
l(`one curve and K_=m or T for the other, and then lets the latter define K.`);
l(`   The curve(s) are specified as follows. Each curve in the present apecs`);
l(`session has a place in the "stack" which consists of the lists ft1,`);
l(`ft2,ft. These contain info. about these curves --- unless stack storage`);
l(`has been toggled off by the command Store() (see Menu(Store)). Thus`);
l(`ft1[n] etc. contain the info. of the curve with stack number n. The args`);
l(`n1 (and n2 if present) must be pos. integer(s) <=nops(ft1). The stack`);
l(`number of the present curve is stac. Thus Isom(stac) or Isom(n1) where`);
l(`n1=stac finds the auto.'s of the present curve. If n1<>stac then Isom(n1)`);
l(`finds the isom.'s of curve #n1 to the present curve. Isom(n1,n2) finds`);
l(`the isom.'s of curve #n1 to curve #n2, auto.'s if n1=n2.`); 
    elif i<156 then
l(`Seekb(args)=NULL        seekb(args)=sBb`);
l(`If K_=1, so the present curve E is defined over Q (see Menu(Ell) for K_),`);
l(`search for rational points on the curves birationally equiv. to E listed`);
l(`in bec with naive height <= args, transfer any points found to E and`);
l(`adjoin to RR when appropriate (as described in Menu(App)), then set`);
l(`sBb:=args. The search starts from the previous sBb (initially sBb:=0).`);
l(`This is occasionally useful when Seek with large args doesn't find a`);
l(`point on E itself; the procedure bas may call seekb.`);
l(`   Whenever a new curve is added to bec sBb is reset to 0.`);
    elif i<158 then
l(`qv is an abbreviation of the Latin quod vide, meaning look there to`);
l(`find out something.`);
    elif i<160 then
l(`Quarm(args), quarm(args)  ---  same args and RETURN values as Quar, quar`);
l(`   These procs call Quar or quar, passing args unchanged, and with`);
l(`quarm_flag set = true, which is a signal to quar to regard the resulting`);
l(`weierstrass equation as being defined over the quadratic field`);
l(`Q(sqrt(m_)) to which apecs is currently pointing. Thus Quarm and quarm`);
l(`require the following two things:`);
l(`   --- Qfin has been previously called so that apecs is pointing to`);
l(`some Q(sqrt(m_)); and`);
l(`   --- args consists of numbers from the field Q(sqrt(m_)).`);
l(`   These procs are principally used when args are rational numbers,`);
l(`since then Quar or quar alone would automatically have quarm_flag=false`);
l(`and would process args in the usual manner over Q.`);
l(`EXAMPLES: `);
l(`   Qfin(3);Quarm(2,0,1,1,3); Apecs notices that [U,V]=[0,w_]`);
l(`(where w_=sqrt(3) --- see Menu(Qfin)) lies on the quartic`);
      print(V^2=U^4+U^2+U+3);
l(`and uses this as the "rational point" to transform to a weier. equation.`);
l(`Another point on the quartic is [0,-w_] and apecs doesn't miss the`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`opportunity to apply trcw(0,-w_) (see Menu(trcw)) which in this example`);
l(`results in a contribution to the list RR of (independent) torsion free`);
l(`points.`);
l(`   Qfin(3);Quar(2,0,1,1,3,0,w_); would have the same result --- quarm`);
l(`isn't needed (but still could be used) since apecs detects the`);
l(`presence of w_ in args and automatically sets quarm_flag=true.`);
l(`   Now consider the same quartic over Q(sqrt(2)):`);
l(`Qfin(2);Quarm(2,0,1,1,3);. Apecs notices that the coeff. of U^4`);
l(`is a square in Q(sqrt(2)), hence the two places at infinity on`);
l(`the quartic are rational (over Q(sqrt(2)) of course), and uses one of`);
l(`them to transform to weier. form. trcw applied to the other place at`);
l(`infinity results in a contribution to RR.`);
l(`   Since the class number of the quad. fields in these examples is 1,`);
l(`one can call Minf to convert the equation into a minimal weier.`);
l(`equation. Note that RR and the torsion subgroup are transformed`);
l(`appropriately, and so are not lost.`);
    elif i<162 then 
l(`Ypecs()=NULL         ypecs()=NULL`);
l(`Zpecs()=NULL         zpecs()=NULL`);
l(`Store the present catalog (which presumably was enlarged in the present`);
l(`apecs session either by adding new curves or improving the info. of`);
l(`existing catalog curves) on the hard disk, and`);
l(`--- in the case of Ypecs or ypecs, return to the present apecs session,`);
l(`--- in the case of Zpecs or zpecs, quit Maple.`);
l(`   What is saved to hard disk are the lists et1, et2, etj and the table`);
l(`eT. The command Nota() provides descriptions of these data.`);
l(`   If you wish to quit without saving the catalog simply type in the`);
l(`Maple command quit or click on File then on Exit. Of course the catalog`);
l(`as it was at the beginning of this session is still on disk, and`);
l(`available for the next session.`);
l(`   Exceptionally you may receive a message that there was trouble`);
l(`along with the value of the list trou, and a request to type in "yes"`);
l(`or "no" --- whether or not to save the catalog anyway. This indicates a`);
l(`programming fault in apecs, and it would help me if you would e-mail`);
l(`the value of trou and, if possible, the input that gave rise to the`);
l(`trouble at`);
print(`connell@math.mcgill.ca`);
    elif i<166 then
l(`Afac(n)=NULL            afac(n)=[u,[[p1,e1],..,[pk,ek]]]`);
l(`n is a nonzero rational whose unique factorization is`);
print(`n = u*p1^e1* .. *pk^ek`);
l(`where u is in {1,-1} and the pi are primes.`);
l(`    afac is like ifactors (but enhanced -- see last paragraph below)`);
l(`with the additional option remember, the point of which is to offer`);
l(`the user the facility of informing apecs of difficult, time consuming`);
l(`factorizations (e.g. of large DD and denom(jay)): at any time in an`);
l(`apecs session one can enter a statement of the form`);
print(`afac(133581487356257425101351408665453450):=[1, [[2, 1], [5, 2], [4\
21, 1], [369137, 1], [35171526301168303, 1], [234653, 1], [2083, 1]]];`);
l(`afac does not check the correctness of the factorization. Also,`);
l(`afac knows the factorization of -n once it knows that of n. The verbose`);
l(`command Afac knows all that afac knows, but afac, not Afac, must be`);
l(`used to inform apecs of factorizations such as the example above.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    To take a simpler (trivial) example, if one enters`);
print(`afac(-3/2):=[-1,[[3,1],[2,-1]]];`);
l(`then any subsequent call to either afac(+-3/2) made by apecs (or the`);
l(`user) will elicit an immediate response --- afac will not have to`);
l(`"sweat through" the factorization. Afac(n) is like afac(n) except it`);
l(`also displays the remember table of afac; Afac() simply displays`);
l(`the table. This table is lost when apecs is quit, so if you expect to`);
l(`encounter some of the same factorizations in future apecs sessions,`);
l(`you could put the required afac(n):=[u,[[..]]] statements in a file,`);
l(`say my_afacs, and then in any session you can enter read my_afacs;,`);
l(`of course after read apecs;.`);
l(`   afac calls afactor which is modelled on Maple's ifactor but which`);
l(`also maintains a set (called afacprimes) of the primes > 1600 that are`);
l(`encountered. Thus if apecs has already factored DD (or -DD), perhaps`);
l(`with the aid of the user by an afac(DD):=... entry, then apecs can`);
l(`quickly factor, e.g., denom(jay) since its prime factors are in`);
l(`the set afacprimes. This feature can be turned off by setting`);
l(`afacflag:=false, or turned back on by afacflag:=true.`);
    else
l(`Heeg(args)=NULL`);
l(`heeg(args)=NULL`);
l(`args (a positive integer) is optional; if present, n:=args, or in`);
l(`default, n:=50000 -- see below`);
l(`For these proc.'s the present curve must have the`);
l(`field label K_=1 (so most likely was introduced by the Ein command)`);
l(`and have (probable) rank r4=1. Heeg checks that K_=1 and at least that`);
l(`the Sign of the Functional Equation SFE=-1, by calling roha if necessary.`);
l(`It also checks that MM (number of indep. points in RR) is <=1.`);
l(`   The canonical height of a Heegner point is calculated; this is`);
l(`available as the (global) variable hht:`);
print(`hht = (approx.) eyy*NN^2/(2*Omega*cP)`);
l(`where eyy is L'(1), the 1st deriv. of the L function as calculated by`);
l(`FnL(4,n) (see Menu(fnL)), NN is the order of the torsion subgroup,`);
l(`Omega is the real period (*2 if the discrim. DD>0 -- see Menu(om)),`);
l(`and cP is the product of the local Tamagawa numbers as calc. in Tate's`);
l(`algorithm. This formula is from Silverman, Computing rational points,..`);
l(`Math. of Comp., 68(1999), 835--858.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`   If the number of terms in the L-series was not sufficient to`);
l(`determine eyy to 4 figures, a suggestion to call Heeg(n+50000) is made.`);
l(`On the other hand, if eyy is deemed to be 0 then Heeg infers that`);
l(`the rank >= 3.`);
l(`   Heeg (but not heeg) displays an upper bound x for the naive height,`);
l(`using hdif_Sik, which suggests calling Seek(x,1) to find the Heegner`);
l(`point.`);
l(`EXAMPLE:         Ein(0,0,0,33,37);Heeg(); ===>`);
l(`Heegner point has canon. height approx. 4.707 and naive ht. < 17200.`);
l(`                 Seek(17200,1); ===>`);
print('RR'=[[924/1849,582173/79507]]);
l(`sB is set = 1848, so a subsequent call to seek will start there.`);
    fi;
  elif member(x,[Clock,clock,'Dat','dat',Half,half,Inch,inch,'menu','\
Nota','nota',Pad,pad,Szp,szp,Tate,tate,Ver,ver]) then
    print(cat(`Menu(`,x,`) not programmed; the brief description given by m\
enu();`));
    l(`should be of some help.`);
  else print(cat(x,` is not an apecs command.`));
  fi;
end:

menu:=proc() local l;
#print list of apecs commands, minimal details
  l:=proc();lprint(args);end:
l(`Note: A capitalized command, e.g. Isog(); gives detailed output, w\
hile`);
l(`the uncapitalized version isog(); gives only bottom line results.`);
l(`In partic., Menu(Ell); e.g., gives more detail than menu(); about Ell`);
l(`Commands accept a point as an argument either as a list [x,y],or s\
equence`);
l(`x,y and also, in the case of a point of finite order n, the form [x,\
y,n].`);
l(`[] denotes point at infinity. In args column, -- stands for \
"nothing"`);
l(`The args "curve" stands for any of: (i) a sequence a1,a2,a3,a4,a\
6 of five`);
l(`rational Weierstrass coefficients, e.g. 0,-2/3,10,0,1, (ii) a cat\
alog`);
l(`name e.g. A11, (iii) a catalog number e.g. 2 (this specifies curv\
e B11).`);
l(`The args "CUR" stands for 5 possibly irrational Weierstrass coeffici\
ents.`);
l(`E stands for the present curve (whose catalog name and number are c\
ur,ncur`);
l(`except that when cur='cur'--i.e. cur has no value--then E was input \
in`);
l(`"general format" and may have irrational coefficients).`); 
print(` `);
l(`Apecs  |args        |description`);
l(`command|            |`);
l(`__________________________________________________________________\
________`);
l(`Abel   | -- or curve| For which p is there an abelian ext. of Q_p \
over`);
l(`       |            | which E acquires good or mult. reduction?`);
l(`Absc   | x          | find point [x,y] on E with given abscissa`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Afac   | n          | =ifactor(n) with option remember`);
l(`AgM    | x,y        | arithmetic-geometric mean`);
l(`Allp   | p          | all points and group structure of E mod p`);
l(`App    | [x,y]      | append [x,y] to RR(=list of points indep. mo\
d torsion)`);
l(`Bas    | -- or BASL | seek("up to BASL") basis RR of E(Q) and regulat\
or reg`);
l(`Cld    | --         | clear denom.'s of Weier. coeff.'s of E/Q`);
l(`Clock  | --         | give status of Clock (uncapital. clock(x) must \
have x)`);
l(`Clock  | x          | allow at most x seconds for infinite series & s\
earches`);
l(`Comb   | n1,z1,n2,  | calc. n1*z1+n2*z2+..+nk*zk any k, any ni in Z,`);
l(`       |   z2,..    |   where each zi=[xi,yi] is a point on E`);
l(`Crem   |0,1 or 2 of A,dil| rank (or upper b.) by Birch+Sw.-Dyer+Cre\
mona alg.`);
l(`Cubic_fit| args     | find general cubic f_=s_1*x_^3+.. satisfying args.`);
l(`       |            | See Menu(Cubic_fit).`);
l(`Dat    | -- or curve| data of E or curve; to display only the`);
l(`       |            | Weierstrass equation use We`);
l(`Div    | n          | calc. al.n as poly in X (essent. the n-divi\
sion poly.`);
l(`Eadd   | z1,z2      | add two points zi=[xi,yi] on E`);
l(`Eadp   | z1,z2      |  "   "    "      "         " " mod p`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Ein    | curve      | use Laska's alg. and Tate's alg. to find min. w\
eier.`);
l(`       |            | form/Z, Kodaira types etc., list all twists i\
n catalog`);
l(`       |            |   also assign catalog name if curve is new`);
l(`Ein0   | a0,..,a6   | as Ein except also coef. a0 of x^3`);
l(`Ell    |curve or CUR| input any ((ir)rational, literal, ..) weier.\
 coef.'s`);
l(`       |&(optional)K_| & (opt.) field label K_. Calc. b's, c's, DD, jay`);
l(`Emod   | p          | count number of points on E mod p (see also A\
llp)`);
l(`Emods  | p1,p2      | for all primes between p1, p2 list Emod, cos t\
heta,`);
l(`       |            |   state if supersing., anomalous, split mult. o\
r ..`);
l(`Exam   | -- or i    | display index for Exam topics/functions or topi\
c #i`);
l(`Exam   | n,t        | input curve with torsion Cn or C2xCn for param.`);
l(`       | or n,t,x   |      t (or t,x); to see details call Exam(1); .`); 
l(`Expr   | z          | express z=n1*P1+n2*P2+..+tor. point where RR=[P\
1,..]`);
l(`Fact   | a+b*w_     | factorize args in Z[w_]; see Menu(Qfin).`);
l(`FnL    | r          | calc. L^{r}(1) (r same parity as sign of func. \
eqn.)`);
l(` "     | r,de       | optional param.: calc. series to within +-10^(-d\
e)`);
l(` "     | r,de,hyy   | optional param.: use at most hyy terms of serie\
s.`);
l(`Ford   | x,y,t      | list multiples of z=[x,y] up to min{t,ord z}`);
l(`Fundunit| --        | smallest unit >1 of real Q(sqrt(m_)); see Menu(Qfin)`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Gcub   | s1,..,sn   | cubic s1*U^3+s2*U^2*V+..+s9*V+s10 with point [s11,\
s12]`);
l(`       | n=10,12 or | or [s11,s12,0] at infinity or search for such`);
l(`       |   13       | provided all args rational, ----> Weierstrass form.`);
l(`Genj   | j          | invoke Ein (Ell for non-rat. j) for "generi\
c j" curve`);
l(`Go     | --         | list stack(=list of curves of this apecs s\
ession)`);
l(`Go     | n or psn   | go to #n in stack or to previous E`);
l(`Half   | z          | q=list of points w such that 2*w=z`);
l(`Hdif_Sik| --        | l. bound lowB for ht(x)-h(x)/2 (h=log. n\
aive height)`);
l(`Hdif_Sil| --        | lowB_Sil,uppB for ht(x)-h(x)/2 (h=log. n\
aive height)`);
l(`Heeg   | -- or n    | Canon. ht. of Heegner pt. using<50000 (or n)`);
l(`       |            | terms in series for L'(1)`);       
l(`Ht     | x,y        | canon. height of [x,y] (Silverman-Tate alg.)`);
l(`Inch   | x          | interchange catalog names (but not numbers)`);
l(`       |            |   of E and curve named x in catalog`);
l(`Isog   | --         | find all curves isogenous to E`);
l(`Lin    | z1,..      | find lin. reln. among zi mod torsion or`);
l(`       |            | find them indep. Also adet:=height pairing`);
l(`       |            | determinant when E and zi are /Q`);
l(`Menu   | x          | help for apecs command x (x stripped of (args\
);`);
l(`       |            |   maple's help(<maple item>); still available`);
l(`menu   | --         | this display`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Mest   | mL         | calc. Mestre's upper bound for rank with param.\
=ln(mL)`);
l(` "     | --         | state result of last call to mest`);
l(`Minf   | --         | min. Weier. form of E/Z[w_]; see Menu(Qfin).`);
l(`Mulp   | N,z        | with p preset, find N*z on E mod p, any N in Z`);
l(`Mult   | N,z        | find N*z on E, any N in Z`);
l(`Neg    | z          | calculate -z`);
l(`Negp   | pz         | negative nz of point pz on E mod p, prime p pr\
eset`);
l(`Nota   | --         | list most important notation used by apecs`);
l(`Om     | --         | lattice periods omega[i], i=1,2 (using AgM)`);
l(`Omp    | n          | p-adic period (Tate's q) and u^2 (Henniart-Me\
stre)`);
l(`       |            |   to n p-adic digits for all relevant p`);
l(`Omp    | p,n        | as Omp(n) but only for specified p`);
l(`On     | x,y        | point on E? - true or false`);
l(`Onp    | x,y        | point on E mod p? - true or false (p preset)`);
l(`Pad    | a,p,n      | p-adic expansion of a mod p^n`);
l(`Qfin   | m          | point apecs to Q(sqrt(m_)); but see Menu(ell)`);
l(`Quar   | S1,..,Sn   | birat. transf. V^2=quartic S1*U^4+..+S5 with \
point`);
l(`       | n=5,7 or 8 |   [U,V]=[S6,S7] or at inf. if n=8 (or poss. search`);
l(`       |            |   for such if nargs=5) to weier. form and invoke`);
l(`       |            |   Ein or Ell depending on K_`);
l(`Quar0  | S1,..      | as Quar but treat Si as irrational even if rat.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Quarm  | S1,..      | with Si in Q call Quar treating Si in Q(sqrt(m_)`);
l(`       |            | Qfin(m_) must be called previously.`);
l(`Raf    | --         | raF=[x-coord.('s) of real point(s) of order 2] o\
r`);
l(`       |            |   x-coord's. of all pts. of order 2 when cur='c\
ur'.`);
l(`Rk     | --         | rank (use Mazur's upper b. or 2-descent if poss\
. or`);
l(`       |            |   parity of rank from roha(), or`);
l(`       |            |   Mestre's upper b. - assume R.H. for L - or`);
l(`       |            |   by eval. of L^{r}(1)) - assuming B-Sw.D)`);
l(`Rk     | d          | as Rk(); but search limit for homog. spaces = d\
`);
l(`Rk0    | -- or d    | as Rk but no Mest() or L^{r}(1)`);
l(`Rk1    | -- or d    | as Rk but no L^{r}(1)`);
l(`RkNC   | -- or d    | as Rk but use of No Conjectures allowed`);
l(`Roha   | --         | SFE as root number.`);
l(`Search | l or l1,l2 | simplif. Seek--no adjust. to naive ht.,no add's\
 to RR`);
l(`Seek   | l          | search for [x,y] up to (modified) naive h\
eight l`);
l(`Seek1  | l or l1,l2 | IP=list integral [x,y] "up to" l or betw. l1 & l2`);
l(`Seekb  | l          | search for points with naive ht<=l on curves in bec`);
l(`Sfe    | --         | sign of functional equation analytically`);
l(` "     | hyy        | " but use at most hyy terms of series for cusp \
form`);
l(`Sha    | -- or X    | order of Shafarevich-Tate group (assuming B-\
Sw.D)`);
l(`Store  | --         | toggle between store/no store curves in cat & s\
tack`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Sub    | z1,z2      | calculate z1-z2`);
l(`Subg   | z1,..      | subgroup of E generated by {z1,..}`);
l(`Szp    | --         | Szpiro ratio (log(abs(DD))/log(Nc))`);
l(`Tate   | --         | Tate's alg. for Neron model (auto. called by e\
in`);
l(`       |            |   so not normally invoked by user)`);
l(`Third_pt| z1,z2     | find 3rd pt. of intersection with cubic f_`);
l(`Tor    | --         | torsion subgroup of E over Q or Q(T) or Q(sqrt(m_)`);
l(` "     | z          | " " with known torsion point z to speed up Tor`);
l(`Torq   | -- or n    | " " /Q(sqrt(m_)), but largely superseded by Tor`);
l(`Trans  | r,s,t,u    | call Ell or ell(a1',..,a6'), a1'=(a1+2s)/u^2, etc.`);
l(`Trcw   | u,v,n      | transfer point [u,v] (eg. []) from curve #n in \
bec`);
l(`  "    | or [u,v],n | (ie. curve as orig. input by Ein,Ein0,Quar\
,Gcub, or`);
l(`  "    |u,v or [u,v]|  Genj) to E, the Weier. form. n absent===>n=1`);
l(`  "    | n or --    | Display curve #n of bec & transf's. n absent===\
>n=1`);
l(`Tri    | z,tric     | image of point z on isog. curve tric(=cat nam\
e or no.)`);
l(`Trwc   | x,y,n  or  | as Trcw except transfer [x,y] on weier. to curv\
e`);
l(`       | [x,y]  etc |   #n in bec`);
l(`Tw     | a          | twist E by a and invoke ell or ein`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`Ub_tor | -- or n    | upper b. for torsion of E/Q(sqrt(m));see Menu(qfin)`);
l(`Velu   | z1,..      | divide E by subgroup generated by {z1,..}, inv\
oke ein`);
l(`       | or poly.   | If args=(X-x1)*.. then subg.={O,+-[x1,y1],..}.\
`);
l(`Ver    | --         | version number, author's address..`);
l(`We     | -- or curve| print Weierstrass equation`);
l(`Weier_fit| args     | find general Weier. eqn. f_=x_^3..-a_3y sat. ar\
gs`);
l(`Ypecs  | --         | as Zpecs(); but don't quit - return to ape\
cs prompt`);
l(`Zpecs  | --         | store updated/enlarged catalog of elliptic cur\
ves`);
l(`       |            |   and data, then quit apecs. To leave apecs wi\
thout`);
l(`       |            |   saving this session's curves+data use maple'\
s quit`);
end: