lprint(`Reading Nota/nota ..`);

Nota:=proc() local l;
#Describe the most important apecs variables.
   l:=proc();lprint(args);end:
l(`The weierstrass coefficients a1,a2,a3,a4,a6 appear in the (weierstrass)`);
l(`equation of an elliptic curve:`);l(``);
print(`Y^2+a1*X*Y+a3*Y = X^3+a2*X^2+a4*X+a6`);
l(`At the beginning of apecs the ai are unevaluated (mathematically they`);
l(`are five independent transcendentals) and the following quantities are`);
l(`defined:`);
l(`    b2:= a1^2+4*a2`);
l(`    b4:= a1*a3+2*a4`);
l(`    b6:= a3^2+4*a6`);
l(`    b8:= a1^2*a6-a1*a3*a4+4*a2*a6+a2*a3^2-a4^2`);
l(`    c4:= b2^2-24*b4`);
l(`    c6:= -b2^3+36*b2*b4-216*b6`);
l(`    DD:=(c4^3-c6^2)/1728 -- the discriminant`);
l(`    jay:=c4^3/DD -- the j-invariant`);
l(`    prac:=x^3+(b2/4)*x^2+(b4/2)*x+b6/4 (roots=x-coord.'s of points of \
order2)`);
l(`    QuarQ:=b2^2/16-2*b4-b2*x/2-3*x^2-(8*y+4*(a1*x+a3))*M-(b2/2+6*x)*M\
^2+M^4`);
l(`          --- see below for the purpose of this poly.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`the Fricke polynomials`);
l(`    fr2:=(F+16)^3-jay*F`);
l(`    fr3:=(F+27)*(F+3)^3-jay*F`);
l(`    fr5:=(F^2+10*F+5)^3-jay*F`);
l(`    fr7:=(F^2+13*F+49)*(F^2+5*F+1)^3-jay*F`);
l(`    fr13:=(F^2+5*F+13)*(F^4+7*F^3+20*F^2+19*F+1)^3-jay*F`);
l(`the division polynomials`);
l(`    al0:=(4*x^3+b2*x^2+2*b4*x+b6)^2`);
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(`At any time these quantities have their values defined according to`);
l(`the currently assigned values of a1,..,a6.`);
l(`fr5,..,fr13 are used in isog: rational roots of fri correspond to`);
l(`rationally defined isogenies of degree i.`);
l(`ali for i>4 are calculated by Div recursively as follows:`);
l(`  (The variable aln is the max. i such that ali has been calculated.\
)`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    al.(4*m+1):=al0*(al.(2*m+2))*(al.(2*m))^3-(al.(2*m-1))*(al.(2*m+\
1))^3`);
l(`    al.(4*m+2):=al.(2*m+1)*((al.(2*m+3))*(al.(2*m))^2-\
(al.(2*m-1))*(al.(2*m+2))^2)`);
l(`    al.(4*m+3):=al.(2*m+3)*(al.(2*m+1))^3-al0*(al.(2*m))*(al.(2*m+\
2))^3`);
l(`    al.(4*m+4):=al.(2*m+2))*((al.(2*m+4))*(al.(2*\
m+1))^2-(al.(2*m))*(al.(2*m+3))^2)`);
l(`These are related to the usual division polynomials (Silverman, p.105)`);
l(`by ali=psi_i for i odd and psi_i/(2y+a1x+a3) for i even except psi_0=\
0.`);
l(`(The value of al0 is a constant used in the above recurrences.)`);
l(`If P=[x,y] is a point on the curve then the rational roots M of QuarQ`);
l(`are in 1--1 correspondence`);
l(`with the solutions Q of [2]Q=P --- used by the procedure Half(P).`);
l(`   The ai are assigned new values, that is, new curves are introduced`);
l(`by one of the apecs commands`);
print(`Ein, Ell, Gcub, Quar, Genj, Cubic_fit, Weier_fit, ...`);
l(`or one of their variants Ein0, Quarm,...; there are also commands`);
l(`Tw, Trans, Velu,... that change the curve to a related curve. See`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`menu() for a list of all apecs commands and for more detail about a`);
l(`particular command xxx call Menu(xxx). In particular, Menu(Ell)`);
l(`explains the field label K_ that is attached to each curve. A curve`);
l(`defined over the rational field Q, i.e., all ai are in Q, is usually`);
l(`introduced by Ein(a1,..,a6) and then K_ is automatically set = 1,`);
l(`and is essentially of no concern. It really is only when one wants to`);
l(`consider curves over the rational function field Q(T) or a quadratic`);
l(`field Q(sqrt(m_) (see Menu(Qfin)) that one must specify K_ = T or`);
l(`m_, respectively and that is usually done with the command Ell.`);
l(`    Apecs keeps track of the curves introduced in a given session `);
l(`by maintaining the stack which is a list of the curves with whatever`);
l(`data has been calculated. The command Go allows one to display the`);
l(`identities of the curves and to move back and forth in the stack.`);
l(`The stack number of the present curve is (the apecs variable) stac,`);
l(`while that of the previous curve is psn. Apecs also maintains a`);
l(`catalog of all K_=1 curves along with pertinent data.`);
l(`New curves with K_=1 along with data that is calculated are added`);
l(`to the catalog. To have these additional curves available in future`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`apecs sessions, and also any new data for curves that were already`);
l(`in the catalog, the enlarged catalog must be stored to disk with a`);
l(`zpecs or ypecs command. Of course often one doesn't want the present`);
l(`session's curves permanently archived, so the session is ended with`);
l(`Maples's quit command. Apecs does not have a mechanism to archive`);
l(`curves with K_<>1.`);
l(`    When the present curve has field label K_=1, the catalog name`);
l(`and number are cur and ncur.`);
l(`    The apecs starter catalog contains all curves/Q of conductor`);
l(`<=124 plus a number of others for various reasons.`);
l(`The catalog contains the data:`);
l(`    et1=[A11,B11,..], a list of the names of all curves in the cat.`);
l(`    et2=[[0,-1,1,0,0],[0,-1,1,-10,-20],..], a list of lists of the`);
l(`        corresponding minimal weierstrass coefficients;`);
l(`    etj=[-4096/11,-122023936/161051,..], a list of the values of ja\
y;`);
l(`    eT[i] where i=1,2,.., a table whose i-th entry contains 22`);
l(`data items for the i-th curve as follows.`);
l(`Note: if the present curve is in the catalog, equivalently K_=1, then`);
l(`    et1[ncur]=cur, et2[ncur]=[a1,..,a6] and etj[ncur]=jay.`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(` n  |eT[ncur][n]| description`);
l(`----|-----------|----------------------------------------------------`);
l(` 1  | [Nc,badp,Nce] | conductor, bad primes, their exponents in Nc`);
l(` 2  | Kod       | sequence of Kodaira types at the bad primes`);
l(` 3  | NN        | order of the torsion subgroup (default -1)`);
l(` 4  | [PP]      | list of nonzero torsion points: a point [x,y] of`);
l(`    |           | order n appears as [x,y,n]`);
l(` 5  | qR        | quality of RR (Menu(Isog) & Menu(Bas) for details)`);
l(` 6  | RR        | list of independent points of infinite order`);
l(` 7  | rc        | provenance of r4 (Menu(Rk) for details)`);
l(` 8  | r4        | rank (provided rc>=0; default -1)`);
l(` 9  | r1        | upper bound for rank -- no conj.'s (default -1)`);
l(` 10 | dill      | limit of search on homog. spaces or, as of ver. 5.5,`);
l(`    |           | when NN is even, possibly`);
l(`    |[RKI,RKJ,dolt]| (see Menu(Rk)) and then dolt[nops(dolt)]=dill`);
l(`continued next screen ...`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(` 11 | nisog     | list of catalog nos. of isogenous curves (if qR>=0)`);
l(` 12 | disog     | list of degrees of isogenies; a negative degree means`);
l(`    |           |   that only the kernel, not the individual points of`);
l(`    |           |   the kernel are defined over Q.`);
l(` 13 | visog     | list of descriptions of kernels of isogenies`);
l(` 14 | reg       | regulator (possibly--see defn. of qR in Menu(Bas);\
 default 1)`);
l(` 15 | omegA     | lattice and p-adic periods (see Menu(om)); in entries`);
l(`    |           | made by ver.<5.8, only real and p-adic periods.`);
l(` 16 | shaf      | order of Shafarevich-Tate group; default=0`);
l(` 17 | SFE       | sign of the functional equation; default=0`);
l(` 18 | tuQ       | data related to visog (see Velu)`);
l(` 19 | sB        | seek() has searched to naive height sB, or as of ve\
r. 2.9`);
l(`    | +-sB      | - means succesive minima precedence for RR; see Men\
u(Bas)`);
l(` 20 | tW        | list of twists (see Tw)`);
l(` 21 | cP        | product of local component numbers -- in B-SD c\
onj., or`);
l(`    |[cP,cplist]| as of ver. 3.6, where cplist=[..,[p,c_p],..] and \
|c_p|`);
l(`    |           | is a factor >1 of cP. (c_p may be entered as -|c_p|`);
l(`    |           | to flag a certain condition).`);
l(` 22 | misc      | table for user data`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l();l(`To display the catalog data for a curve use Dat or dat.`);
l(`Many curves in the starter catalog have default entries for some of`);
l(`the above entries, such as eT[i][3]=-1 (Tor has not been done),`);
l(`eT[i][8]=-1 (Rk has not been called, or one of the less ambitious`);
l(`rank procedures such as RkNC was not successful), etc. Thus one can`);
l(`see apecs in action by Ein(i);Tor();RkNC();, etc. Cf. the discussion`);
l(`in Exam(4).`);
l(`    In the case of catalog curves (K_=1) the stac contains certain`);
l(`non-catalog data of a more transitory nature,`);
l(`e.g., the table eae[i] of L-series (Fourier) coefficients, i=1..eai.`);
l(`For non-catalog curves, stack contains all that apecs knows about`);
l(`these curves. The stack consists of 3 lists: ft1,ft2 and ft.`);
l(`   ft1[n]=cat. name or "CUR" if K_<>1`);
l(`   ft2[n]=[a1,a2,a3,a4,a6]  when K_=1, or`);
l(`          [a1,a2,a3,a4,a6,K_]  when K_<>1`);
l(`   ft[n]=list of data`);
l(`>>>>>HIT SEMICOLON KEY ;<enter> FOR NEXT SCREEN`);readstat();
l(`    To speed up the determination of the number of points N_p on`);
l(`a catalog curve mod p,`);
l(`apecs maintains tables of the trace of the Frobenius morphism:`);
l(`froB0, tFr, tFr2 and tFr3. For p in the ranges (resp.) 5..97,`);
l(`101..853, 857..1327, 1361..tFr2[1], the p-th entry is a list ---`);
l(`call it X --- such that if X[modp(jay,p)]=b and N_p=p+1-a then`);
l(`a=b*L(c6,p), L denoting the Legendre symbol, except for necessary`);
l(`modifications when jay=0 or 1728 mod p, or p<17. tFr3[1] contains`);
l(`the smallest prime not (yet) in tFr3.`);
l(`(I used ubasic and upecs to construct the Frobenius tables ---`);
l(`much faster than apecs.)`);
l(`    Information about more specialized notation can sometimes be`);
l(`obtained using the Menu command. E.g. Menu(Qfin) explains the meaning`);
l(`of m_, w_, ram_, etc. that are used in the quadratic field package.`);
end:

nota:=proc() local l;
  l:=proc();lprint(args);end:
print(`et1[ncur]=cur, et2[ncur]=[a1,..,a6] and etj[ncur]=jay.`);
l(`eT[ncur]=[Nc*,Kod,NN,PP,qR,RR,rc,r4,r1,dill*,nisog,disog,visog,reg,omeg\
a,shaf,SFE,tuQ,sB*,tW,cP*,misc]`);
l(`* the first entry is Nc or, as of version 2.00 of apecs,`);
print(`Ncl=[Nc,badp,Nce]`);
l(`* sB may have - flag attached as of version 2.9.`);
l(`* cP may be replaced by [cP,cplist] as of version 3.6.`);
l(`* dill may be replaced by [RKI,RKJ,dolt] as of version 5.5.`);
l(`For more details use the capitalized command Nota();.`);
end: