Q0 = [0,0] [43A-5,1] = 2 Q0 [43A-5,2] = [] [43A-8,1] = -2 Q0 [43A-8,2] = [] [43A-20,1] = -4 Q0 [43A-20,2] = [] [43A-29,1] = 2 Q0 [43A-29,2] = [] [43A-37,1] = 2 Q0 [43A-37,2] = [] [43A-61,1] = -2 Q0 [43A-61,2] = [] [43A-73,1] = -2 Q0 [43A-73,2] = [] [43A-89,1] = 2 Q0 [43A-89,2] = [] [43A-113,1] = 4 Q0 [43A-113,2] = [] [43A-116,1] = -4 Q0 [43A-116,2] = [] [43A-125,1] = -10 Q0 [43A-125,2] = [] [43A-137,1] = 4 Q0 [43A-137,2] = [] [43A-149,1] = [] [43A-149,2] = [] [43A-157,1] = -4 Q0 [43A-157,2] = [] [43A-12,1] = -1 Q0 [43A-12,2] = -1 Q0 [43A-12,3] = [-5/4, -1/2+3/8*I] [43A-12,4] = [-5/4, -1/2-3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-28,1] = 1 Q0 [43A-28,2] = 1 Q0 [43A-28,3] = [-5/4, -1/2+3/8*I] [43A-28,4] = [-5/4, -1/2-3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-32,1] = 3 Q0 [43A-32,3] = 3 Q0 [43A-32,2] = [-5/4, -1/2-3/8*I] [43A-32,4] = [-5/4, -1/2+3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-33,1] = 1 Q0 [43A-33,2] = 1 Q0 [43A-33,3] = [-141/44, -1/2-1381/968*I*11^(1/2)] [43A-33,4] = [-141/44, -1/2+1381/968*I*11^(1/2)] \brak{-\frac{141}{44}, -\frac{1}{2} \pm \frac{1381}{968} \sqrt{-11}} [43A-45,1] = -2 Q0 [43A-45,3] = -2 Q0 [43A-45,2] = ???? [43A-45,4] = - [43A-45,2] [43A-48,1] = 3 Q0 [43A-48,2] = 3 Q0 [43A-48,3] = [-5/4, -1/2-3/8*I] [43A-48,4] = [-5/4, -1/2-3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-65,1] = -1 Q0 + [61/52, -1/2-675/1352*13^(1/2)] [43A-65,2] = -1 Q0 + [61/52, -1/2+675/1352*13^(1/2)] -1 P_0 + \brak{\frac{61}{52}, -\frac{1}{2} \pm \frac{675}{1352} \sqrt{13}} [43A-65,3] = [] [43A-65,4] = [] [43A-69,1] = 1 Q0 [43A-69,2] = 1 Q0 [43A-69,3] = [-36/23, -1/2+235/1058*I*23^(1/2)] [43A-69,4] = [-36/23, -1/2-235/1058*I*23^(1/2)] \brak{-\frac{36}{23}, -\frac{1}{2} \pm \frac{235}{1058} \sqrt{-23}} [43A-72,1] = 2 Q0 [43A-72,2] = 2 Q0 [43A-72,3] = ???? [43A-72,4] = - [43A-72,3] [43A-76,1] = [] [43A-76,2] = [] [43A-76,3] = 2 [-5/4, -1/2+3/8*I] [43A-76,4] = 2 [-5/4, -1/2-3/8*I] 2 \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-77,1] = -3 Q0 [43A-77,2] = -3 Q0 [43A-77,3] = [-141/44, -1/2-1381/968*I*11^(1/2)] [43A-77,4] = [-141/44, -1/2+1381/968*I*11^(1/2)] \brak{-\frac{141}{44}, -\frac{1}{2} \pm \frac{1381}{968} \sqrt{-11}} [43A-80,1] = 1 Q0 [43A-80,2] = 1 Q0 [43A-80,3] = [-5/4, -1/2+3/8*I] [43A-80,4] = [-5/4, -1/2-3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-85,1] = -1 Q0 + [-19/17, -1/2-45/578*17^(1/2)] [43A-85,3] = -1 Q0 + [-19/17, -1/2+45/578*17^(1/2)] -1 P_0 + \brak{-\frac{19}{17}, -\frac{1}{2} \pm \frac{45}{578} \sqrt{17}} [43A-85,2] = [] [43A-85,4] = [] [43A-88,1] = 1 Q0 [43A-88,2] = 1 Q0 [43A-88,3] = [-141/44, -1/2-1381/968*I*11^(1/2)] [43A-88,4] = [-141/44, -1/2+1381/968*I*11^(1/2)] \brak{-\frac{141}{44}, -\frac{1}{2} \pm \frac{1381}{968} \sqrt{-11}} [43A-93,1] = 3 Q0 [43A-93,2] = 3 Q0 [43A-93,3] = [-392/31, -1/2+14895/1922*I*31^(1/2)] [43A-93,4] = [-392/31, -1/2-14895/1922*I*31^(1/2)] \brak{-\frac{392}{31}, -\frac{1}{2} \pm \frac{14895}{1922} \sqrt{-31}} [43A-104,1] = 1 Q0 + [61/52, -1/2+675/1352*13^(1/2)] [43A-104,3] = 1 Q0 + [61/52, -1/2-675/1352*13^(1/2)] 1 P_0 + \brak{\frac{61}{52}, -\frac{1}{2} \pm \frac{675}{1352} \sqrt{13}} [43A-104,2] = [] [43A-104,4] = [] [43A-108,1] = 3 Q0 [43A-108,2] = 3 Q0 [43A-108,3] = [-5/4, -1/2-3/8*I] [43A-108,4] = [-5/4, -1/2+3/8*I] \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-112,1] = -3 Q0 [43A-112,2] = -3 Q0 [43A-112,3] = 3 [-5/4, -1/2+3/8*I] [43A-112,4] = 3 [-5/4, -1/2-3/8*I] 3 \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-128,1] = -4 Q0 [43A-128,3] = -4 Q0 [43A-128,2] = 2 [-5/4, -1/2+3/8*I] [43A-128,4] = 2 [-5/4, -1/2-3/8*I] 2 \brak{-\frac{5}{4}, -\frac{1}{2} \pm \frac{3}{8} \sqrt{-1}} [43A-132,1] = -4 Q0 [43A-132,2] = -4 Q0 [43A-132,3] = [] [43A-132,4] = [] [43A-141,1] = -2 Q0 [43A-141,2] = -2 Q0 [43A-141,3] = 2 [-7, -1/2+5/2*I*47^(1/2)] [43A-141,4] = 2 [-7, -1/2-5/2*I*47^(1/2)] 2 \brak{-7, -\frac{1}{2} \pm \frac{5}{2} \sqrt{-47}} [43A-161,1] = 1 Q0 [43A-161,2] = 1 Q0 [43A-161,3] = [-36/23, -1/2+235/1058*I*23^(1/2)] [43A-161,4] = [-36/23, -1/2+235/1058*I*23^(1/2)] \brak{-\frac{36}{23}, -\frac{1}{2} \pm \frac{235}{1058} \sqrt{-23}} [43A-177,1] = -2 Q0 [43A-177,2] = -2 Q0 [43A-177,3] = 2 [-19/16, -1/2-1/64*I*59^(1/2)] [43A-177,4] = 2 [-19/16, -1/2+1/64*I*59^(1/2)] 2 \brak{-\frac{19}{16}, -\frac{1}{2} \pm \frac{1}{64} \sqrt{-59}} [43A-180,1] = 4 Q0 [43A-180,2] = 4 Q0 [43A-180,3] = ???? [43A-180,4] = - [43A-180,3] [43A-184,1] = -1 Q0 [43A-184,2] = -1 Q0 [43A-184,3] = [-36/23, -1/2-235/1058*I*23^(1/2)] [43A-184,4] = [-36/23, -1/2+235/1058*I*23^(1/2)] \brak{-\frac{36}{23}, -\frac{1}{2} \pm \frac{235}{1058} \sqrt{-23}} [43A-200,1] + [43A-200,2] = 8 Q0 [43A-200,1] - [43A-200,2] = 4 [1/2, -1/2-1/4*10^(1/2)] [43A-200,1] = 4 Q0 + 2 [1/2, -1/2-1/4*10^(1/2)] [43A-200,2] = 4 Q0 + 2 [1/2, -1/2+1/4*10^(1/2)] 4 P_0 + 2 \brak{\frac{1}{2}, - \frac{1}{2} \pm \frac{1}{4} \sqrt{10}} [43A-200,3] = [] [43A-200,4] = [] for(j=1,200,trap(,;,if(kronecker(j,43)==-1&&length(qfbclassgroup(j))==2,print(j);print(hpxhgetpoint1([43,A],3,j,[1]));print(hpxhgetpoint1([43,A],3,j,[2]))))) for(j=1,200,trap(,;,if(kronecker(j,43)==-1&&length(qfbclassgroup(j))==2,print(Str("[43A-"j",1] = \n[43A-"j",2] = \n[43A-"j",3] = \n[43A-"j",4] = \n"))))) CLASS NUMBER 3 EXAMPLE!!! [1,1,1,1] -> [] [1,-1,-1,1] -> 4 [1/4, -1/2+1/8*21^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> [] [43A-105,1] = [1/4, -1/2+1/8*21^(1/2)] [43A-105,2] = [1/4, -1/2-1/8*21^(1/2)] [43A-105,3] = [1/4, -1/2-1/8*21^(1/2)] [43A-105,4] = [1/4, -1/2+1/8*21^(1/2)] \brak{\frac{1}{4}, -\frac{1}{2} \pm \frac{1}{8} \sqrt{21}} --- [1,1,1,1] -> [] [1,-1,-1,1] -> [] [1,1,-1,-1] -> 4 [-2, -1/2-1/2*I*15^(1/2)] [1,-1,1,-1] -> 4 [-47/36, -1/2+19/216*I*35^(1/2)] [43A-105,5] = [-2, -1/2-1/2*I*15^(1/2)] + [-47/36, -1/2+19/216*I*35^(1/2)] [43A-105,6] = [-2, -1/2-1/2*I*15^(1/2)] + [-47/36, -1/2-19/216*I*35^(1/2)] [43A-105,7] = [-2, -1/2+1/2*I*15^(1/2)] + [-47/36, -1/2+19/216*I*35^(1/2)] [43A-105,8] = [-2, -1/2+1/2*I*15^(1/2)] + [-47/36, -1/2-19/216*I*35^(1/2)] \brak{-2, -\frac{1}{2}+\frac{1}{2} \sqrt{-15}} + \brak{-\frac{47}{36}, -\frac{1}{2} + \frac{19}{216} \sqrt{-35}} and conjugates [1,1,1,1] -> [] [1,-1,-1,1] -> 4 [-1/2, -1/2-1/4*6^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> [] [43A-120,1] = [-1/2, -1/2-1/4*6^(1/2)] [43A-120,2] = [-1/2, -1/2+1/4*6^(1/2)] [43A-120,3] = [-1/2, -1/2+1/4*6^(1/2)] [43A-120,4] = [-1/2, -1/2-1/4*6^(1/2)] \brak{ -\frac{1}{2}, - \frac{1}{2} \pm \frac{1}{4} \sqrt{6} } --- [1,1,1,1] -> [] [1,-1,-1,1] -> 4 [-209/162, -1/2+445/2916*I*10^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> 4 [-2, -1/2+1/2*I*15^(1/2)] [43A-120,5] = [-209/162, -1/2+445/2916*I*10^(1/2)] + [-2, -1/2+1/2*I*15^(1/2)] [43A-120,6] = [-209/162, -1/2-445/2916*I*10^(1/2)] + [-2, -1/2-1/2*I*15^(1/2)] [43A-120,7] = [-209/162, -1/2-445/2916*I*10^(1/2)] + [-2, -1/2+1/2*I*15^(1/2)] [43A-120,8] = [-209/162, -1/2+445/2916*I*10^(1/2)] + [-2, -1/2-1/2*I*15^(1/2)] \brak{-\frac{209}{162}, -\frac{1}{2} + \frac{445}{2916} \sqrt{-10}} + \brak{-2, -\frac{1}{2} + \frac{1}{2} \sqrt{-15}} and conjugates [1,1,1,1] -> -2 Q0 [1,-1,-1,1] -> [] [1,1,-1,-1] -> [] [1,-1,1,-1] -> 2 [-19/17, -1/2+45/578*17^(1/2)] [43A-136,1] = [3/2+1/2*17^(1/2), 3+17^(1/2)] [43A-136,2] = [3/2-1/2*17^(1/2), 3-17^(1/2)] [43A-136,3] = [3/2+1/2*17^(1/2), 3+17^(1/2)] [43A-136,4] = [3/2-1/2*17^(1/2), 3-17^(1/2)] \brak{\frac{3}{2} \pm \frac{1}{2} \sqrt{17}, 3 \pm \sqrt{17}} --- [1,1,1,1] -> [] [1,-1,-1,1] -> ???? [1,1,-1,-1] -> ???? [1,-1,1,-1] -> [] [43A-136,5] = [43A-136,6] = [43A-136,7] = [43A-136,8] = [1,1,1,1] -> 2 Q0 [1,-1,-1,1] -> [] [1,1,-1,-1] -> 2 [61/52, -1/2-675/1352*13^(1/2)] [1,-1,1,-1] -> [] [43A-156,1] = [4+13^(1/2), 11+3*13^(1/2)] [43A-156,2] = [4+13^(1/2), 11+3*13^(1/2)] [43A-156,3] = [4-13^(1/2), 11-3*13^(1/2)] [43A-156,4] = [4-13^(1/2), 11-3*13^(1/2)] \brak{4 \pm \sqrt{13}, 11 \pm 3 \sqrt{13}} --- [1,1,1,1] -> [] [1,-1,-1,1] -> 2 [-90725703092354261/21273352283136148, -1/2+23859297904275188235469965/11187307465644197896823464*I*13^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> 2 [-5/4, -1/2-3/8*I] [43A-156,5] = [-152233963/56647368+20226293/56647368*13^(1/2), -1/2+285199304263/75369323124*I-127648135123/150738646248*I*13^(1/2)] [43A-156,6] = [-152233963/56647368+20226293/56647368*13^(1/2), -1/2-285199304263/75369323124*I+127648135123/150738646248*I*13^(1/2)] [43A-156,7] = [-152233963/56647368-20226293/56647368*13^(1/2), -1/2+285199304263/75369323124*I+127648135123/150738646248*I*13^(1/2)] [43A-156,8] = [-152233963/56647368-20226293/56647368*13^(1/2), -1/2-285199304263/75369323124*I-127648135123/150738646248*I*13^(1/2)] \brak{-\frac{152233963}{56647368}-\frac{20226293}{56647368} \sqrt{13}, -\frac{1}{2} - \frac{285199304263}{75369323124} \sqrt{-1} - \frac{127648135123}{150738646248} \sqrt{-13}} and conjugates [1,1,1,1] -> -4 Q0 [1,-1,-1,1] -> [] [1,1,-1,-1] -> 4 [1/4, -1/2+1/8*21^(1/2)] [1,-1,1,-1] -> [] [43A-168,1] = -1 Q0 + [1/4, -1/2+1/8*21^(1/2)] [43A-168,2] = -1 Q0 + [1/4, -1/2+1/8*21^(1/2)] [43A-168,3] = -1 Q0 + [1/4, -1/2-1/8*21^(1/2)] [43A-168,4] = -1 Q0 + [1/4, -1/2-1/8*21^(1/2)] -1 P_0 + \brack{\frac{1}{4}, -\frac{1}{2} - \frac{1}{8} \sqrt{21}} --- [1,1,1,1] -> [] [1,-1,-1,1] -> 4 [-3/2, -1/2+1/4*I*14^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> 4 [-7/2, -1/2-9/4*I*6^(1/2)] [43A-168,5] = [-3/2, -1/2+1/4*I*14^(1/2)] + [-7/2, -1/2-9/4*I*6^(1/2)] [43A-168,6] = [-3/2, -1/2-1/4*I*14^(1/2)] + [-7/2, -1/2+9/4*I*6^(1/2)] [43A-168,7] = [-3/2, -1/2-1/4*I*14^(1/2)] + [-7/2, -1/2-9/4*I*6^(1/2)] [43A-168,8] = [-3/2, -1/2+1/4*I*14^(1/2)] + [-7/2, -1/2+9/4*I*6^(1/2)] \brak{-\frac{3}{2}, -\frac{1}{2} + \frac{1}{4} \sqrt{-14}} + \brak{-\frac{7}{2}, -\frac{1}{2} + \frac{9}{4} \sqrt{-6}} MISSING DATA! [1,1,1,1] -> [1,-1,-1,1] -> [1,1,-1,-1] -> [1,-1,1,-1] -> [43A-192,1] + [43A-192,2] = -4 Q0 [43A-192,1] - [43A-192,2] = 2 [-1/2, -1/2-1/4*6^(1/2)] [43A-192,1] = -2 Q0 + [-1/2, -1/2-1/4*6^(1/2)] [43A-192,2] = -2 Q0 + [-1/2, -1/2+1/4*6^(1/2)] [43A-192,4] = -2 Q0 + [-1/2, -1/2+1/4*6^(1/2)] -2 P_0 + \brak{-\frac{1}{2}, -\frac{1}{2} \pm \frac{1}{4}\sqrt{6}} --- [1,1,1,1] -> [] [1,-1,-1,1] -> 4 [-7/2, -1/2-9/4*I*6^(1/2)] [1,1,-1,-1] -> [] [1,-1,1,-1] -> [] [43A-192,3] = [-7/2, -1/2-9/4*I*6^(1/2)] [43A-192,5] = [-7/2, -1/2+9/4*I*6^(1/2)] [43A-192,6] = [-7/2, -1/2+9/4*I*6^(1/2)] [43A-192,7] = [-7/2, -1/2-9/4*I*6^(1/2)] \brak{-\frac{7}{2}, -\frac{1}{2} \pm \frac{9}{4} \sqrt{-6}}