
// The list of elliptic curves in Theorem 1.1.

shalist :=
[ "3364c", "6552y", "6622b", "7139a", "9450p", "9510e", "10800y", "11970o",
"12927e", "13432b", "13673a", "14938n", "15219c", "17271g", "17816c", 
"18513b", "18550c", "18832a", "19095a", "19350q", "20380b", "20502ba", 
"20544v", "21312ce", "21696l", "21708b", "21771b", "23025c", "23085h", 
"23232dv", "23744v", "24432j", "25296a", "26166h", "26600m", "26640bu", 
"27466a", "27930bj", "28304b", "28314bn", "29025k", "29400di", "29445c", 
"30276r", "30704a", "30996b", "31311h", "31800e", "34626a", "34974h", 
"35682k", "35847h", "36270l", "36300br", "36414u", "38280n", "38316a", 
"39184i", "39975c", "39975o", "39984bf", "40362s", "40598a", "40764a", 
"40950u", "41213a", "41616l", "41616cw", "42218a", "42565b", "43296g", 
"43350q", "43806h", "44115c", "45474i", "45494e", "45738ca", "45770c", 
"46300c", "46389c", "46443c", "46704k", "46800ew", "47232bu", "47294b", 
"47558a", "48195h", "48225l", "49267b", "49300c", "49575b", "49938i", 
"50648b", "51600m", "51912s", "52928c", "53064d", "53865d", "53878g", 
"54045c", "54327c", "54379a", "54450gm", "54663a", "55062o", "55778b", 
"56074a", "56238c", "56510c", "57150g", "57553a", "58029b", "58665a", 
"59536d", "59675n", "60208p", "60885r", "61200gv", "61320v", "61360p", 
"61950q", "62590b", "63518f", "64416f", "65065i", "65088be", "65175t", 
"66352i", "67032cf", "67080r", "67179c", "68170b", "68805d", "69440z", 
"70266j", "70950bc", "71148bh", "71247b", "72128j", "72324j", "72384o", 
"73296bf", "73416g", "73644l", "73815d", "74178c", "74400h", "75075f", 
"75712bz", "76176cd", "76362r", "76608bl", "76755g", "77568s", "78210k", 
"78400dg", "78400le", "78652b", "79013a", "79350cy", "79475p", "79530a", 
"80091b", "80852a", "81498x", "81600ho", "81650n", "81685e", "83050d", 
"83056d", "84042m", "84150gk", "84474w", "84640u", "84672kg", "84681e", 
"84915c", "84966ea", "85050s", "85410f", "86612a", "86700ba", "87362bg", 
"87856n", "88134i", "88305h", "88450f", "89211c", "89425i", "89460d", 
"89656c", "89892c", "90650i", "90950i", "90950m", "91035c", "91800be", 
"92778r", "92862j", "92950be", "93200n", "93396c", "93795e", "94725g", 
"94758g", "94896ba", "95040cd", "95325h", "95590c", "96300bd", "96558i", 
"96746j", "96846c", "96976b", "97470bp", "97482d", "98163a", "98325x", 
"98397i", "98610l", "99586c", "99862a" ];

// Table 1 : Pairs of 7-congruent elliptic curves.

table1 := [ 
[ "3364c", "10092c" ], [ "6552y", "6552ba" ], [ "9450p", "9450t" ], 
[ "9510e", "561090*" ], [ "10800y", "10800u" ], [ "11970o", "11970s" ], 
[ "12927e", "12927d" ], [ "18832a", "1712d" ], [ "19350q", "19350s" ], 
[ "20544v", "20544u" ], [ "21312ce", "21312cd" ], [ "21696l", "21696k" ], 
[ "23232dv", "23232dt" ], ["26600m", "26600l" ], [ "26640bu", "26640bt" ], 
[ "27930bj", "27930bh" ], [ "28314bn", "28314bp" ], [ "29400di", "29400do" ], 
[ "30276r", "30276q" ], [ "30996b", "30996a" ], [ "31311h", "31311k" ], 
[ "31800e", "31800f" ], [ "34974h", "174870bi" ], [ "35682k", "35682j" ], 
[ "36270l", "36270j" ], [ "36300br", "36300by" ], [ "36414u", "36414y" ], 
[ "40362s", "40362t" ], [ "41616cw", "41616cx" ], [ "43296g", "43296f" ], 
[ "43350q", "43350p" ], [ "45494e", "45494d" ], [ "45738ca", "45738bz" ], 
[ "46704k", "6672i" ], [ "46800ew", "46800fa" ], [ "47232bu", "47232bp" ], 
[ "49938i", "49938h" ], [ "51600m", "51600l" ], [ "54450gm", "54450go" ], 
[ "57150g", "57150e" ], [ "60885r", "60885q" ], [ "61200gv", "61200gu" ], 
[ "61950q", "61950o" ], [ "65088be", "65088bc" ], [ "68805d", "68805c" ], 
[ "69440z", "69440ba" ], [ "71148bh", "71148bg" ], [ "72384o", "72384n" ], 
[ "73416g", "10488b" ], [ "74400h", "74400j" ], [ "75075f", "75075d" ], 
[ "75712bz", "10816bf" ], [ "76176cd", "76176cg" ], [ "76362r", "76362q" ], 
[ "76608bl", "3140928*" ], [ "78210k", "7110h" ], [ "78400dg", "235200eb" ], 
[ "79350cy", "79350cx" ], [ "79530a", "15906c" ], [ "81600ho", "81600hn" ], 
[ "84150gk", "84150gl" ], [ "84474w", "84474u" ], [ "84672kg", "84672kd" ], 
[ "84681e", "84681h" ], [ "84966ea", "84966dx" ], [ "85050s", "85050x" ], 
[ "88305h", "88305g" ], [ "88450f", "2967232150*" ], [ "89211c", "681c" ], 
[ "91035c", "18207a" ], [ "91800be", "91800bd" ], [ "92778r", "92778t" ], 
[ "92862j", "13266e" ], [ "92950be", "92950bd" ], [ "93795e", "6659445*" ], 
[ "95040cd", "95040ci" ], [ "96558i", "5082d" ], [ "97470bp", "97470br" ], 
[ "98325x", "25859475*" ] ];

// The elliptic curves in Table 1 beyond the range of Cremona's tables.

FF := [ < 561090, [ 1, 1, 0, 95243, -147561011] >,
 < 3140928, [ 0, 0, 0, -8580204, 8146637424 ]>,
 < 2967232150, [ 1, 0, 1, -3223101295946, 3747757318724534268 ]>,
 < 6659445, [ 0, -1, 1, -67728609901, 7329794977161867 ]>,
 < 25859475, [ 0, 0, 1, -31493068950, -6675516099954594 ]>];

// Table 2 : Hecke eigenvalues a_p(f)

Qrt2<a> := QuadraticField(2);
table2 := [
<"6622b",[ -1, -a - 1, a - 2, -1, 1, 2*a - 2, -3*a - 2, -a - 6, 
  -a - 6, a - 9, -2*a - 6, 5*a - 2, 0, -1, 2*a + 6, 4*a - 3, 
  -3*a - 2, 2*a + 1, -2*a - 4, -4*a - 6, -6*a +  3, -3*a - 9, 
  10*a + 3, 8*a - 4, a - 5 ]>,
<"9510e1",[ -1, -1, 1, a - 3, -2*a - 2, -5, 2*a - 5, -6, -5*a - 1, 
  4*a - 2, -2*a - 2, 4*a - 4, 4*a, -6, -6*a - 2, -4*a - 4, -4, 
  2*a - 9, 4*a - 6, a - 9, -4*a + 2, 8*a, 5*a + 3, -10*a + 3, 4*a - 3 ]>,
<"14938n",[ 1, -2, a - 2, -1, -1, -2, -a - 6, -a - 4, -3*a - 2, 
  -6*a - 2, 2*a - 6, 3*a - 4, 3*a + 2, -4*a + 4, 2*a - 10, 
  4*a - 6, -6*a - 4, 2*a - 12, 8*a - 4, 7*a + 6, 6*a, -2*a - 10,
  -a - 12, -6*a - 4, -1 ]>,
<"15219c",[ -a - 1, 0, -1, 2*a - 2, -5, -2*a + 3, -4*a - 1, -1, 
  2*a - 4, -2*a - 5, -2*a - 7, 2*a + 3, 2*a - 1, 4*a + 2, 8*a, 
  2*a, -2*a - 9, -2*a - 10, 4*a - 2, -4*a - 4, -15, 4*a - 4, 
  -4*a - 10, -1, 2*a + 2 ]>, 
<"20502ba",[ 1, 0, -a - 3, -a - 3, 3*a - 2, a - 2, -1, -6, -5, 
  3*a - 2, -a - 7, -2*a - 1, -6*a - 3, 3*a - 1, -4*a - 4, 4*a - 7, 
  -4*a - 5, 3*a - 10, 1, -8*a - 2, -5*a - 9, 2, 4*a + 7, 
  -2*a + 8, -6*a + 8 ]>
];

// Table 3 : Genus 2 curves
// This data is included in Table 4 below

// Table 4 : Pairs of elliptic curves and genus 2 curves.

// One change from the paper : The conductor of each genus 2 
// Jacobian is (kappa*N)^2 where N is the conductor of E, 
// and kappa is recorded in the second column of the table.

P<x> := PolynomialRing(Rationals());
table4 := [
    <"6622b", 1, [ 41/4, 1/5, 0, 1 ], [ 1, 4 ], [ 11 ], "m", 
20*x^6 + 44*x^5 - 23*x^4 - 10*x^3 + 81*x^2 - 52*x + 4>,
    <"9510e", 1, [ 29/5, 18, 53, 2 ], [ 1, 4 ], [], "g", 
9*x^6 - 12*x^5 + 2*x^4 - 84*x^3 + 437*x^2 + 528*x + 144>,
    <"14938n", 1, [ 9, 1/4, 3, 1 ], [ 2, 4 ], [], "m", 
4*x^6 + 36*x^5 + 37*x^4 - 150*x^3 - 47*x^2 - 72*x + 16>,
    <"15219c", 1, [ 9/8, -4/7, -6/7, 1 ], [ 1, 4 ], [], "g", 
-7*x^6 + 18*x^5 - 93*x^4 + 18*x^3 + 117*x^2 - 36*x + 64>,
    <"20502ba", 1, [ 6, 9/16, 7/4, 1 ], [ 1, 4 ], [], "g", 
36*x^6 + 108*x^5 + 381*x^4 + 474*x^3 + 849*x^2 + 504*x + 144>,
    <"21771b", 1, [ 81/8, 2, 12, 2 ], [ 1, 4 ], [], "g", 
4*x^6 + 36*x^5 - 87*x^4 - 468*x^3 + 657*x^2 + 648*x + 128>,
    <"23025c", 19, [ -52/3, -3/4, 3, 5 ], [ 2, 2 ], [ 19 ], "g", 
-15*x^6 - 510*x^5 - 3865*x^4 + 7750*x^3 - 1555*x^2 - 1560*x + 180>,
    <"23085h", 1, [ 6, 3/5, 22/5, 1 ], [ 1, 4 ], [], "g", 
9*x^6 - 54*x^5 + 27*x^4 - 438*x^3 + 261*x^2 + 360*x + 60>,
    <"23744v", 1, [ -7, -2, 9, 1 ], [ 1, 4 ], [], "m", 
4*x^6 + 4*x^5 - 6*x^4 - 16*x^3 + 22*x^2 - 8*x - 8>,
    <"24432j", 1, [ -8, -3/4, 7, 15 ], [ 1, 4 ], [], "g", 
-5*x^6 + 90*x^5 - 295*x^4 + 890*x^3 - 265*x^2 - 240*x + 60>,
    <"25296a", 37, [ 9, 2/5, 5/2, 1 ], [ 1, 6 ], [ 31, 37 ], "g", 
16*x^6 + 144*x^5 + 323*x^4 - 189*x^3 + 263*x^2 - 48*x + 160>,
    <"26166h", 1, [ 83/12, 4, 95/6, 1 ], [ 1, 4 ], [ 3 ], "g^{(3)}", 
30*x^6 - 174*x^5 + 235*x^4 - 96*x^3 + 85*x^2 + 396*x + 100>,
    <"29445c", 1, [ -14/3, -1/2, 5/2, 1 ], [ 1, 4 ], [], "g", 
9*x^6 - 12*x^5 - 8*x^4 - 74*x^3 + 28*x^2 - 15>,
    <"39984bf", 1, [ 3, 4, 11, 1 ], [ 1, 4 ], [], "g^{(3)}", 
4*x^6 + 12*x^5 + 71*x^4 + 150*x^3 + 191*x^2 - 36*x + 4>,
    <"40764a", 1/3, [ 8, 2, 7, 1 ], [ 1, 4 ], [], "g", 
x^6 + 2*x^5 + 11*x^4 - 6*x^3 - 5*x^2 - 10*x + 9>,
    <"40950u", 1, [ 9, 4/5, 29/5, 5 ], [ 4, 4 ], [], "m", 
20*x^6 + 180*x^5 + 225*x^4 - 650*x^3 - 135*x^2 - 300*x + 100>,
    <"41616l", 1, [ 23/3, 9, 40, 1 ], [ 1, 4 ], [], "g", 
9*x^6 + 15*x^5 - 42*x^4 - 25*x^3 + 123*x^2 - 60*x - 16>,
    <"45770c", 1/2, [ 4, 1, -1, 1 ], [ 1, 4 ], [ 5 ], "g", 
x^6 + 4*x^5 + 32*x^4 - 8*x^3 + 128*x^2 + 44*x + 4>,
    <"54327c", 1, [ 47/7, 7/6, 4, 42 ], [ 2, 4 ], [], "m", 
x^6 + 44*x^4 + 46*x^3 + 56*x^2 + 12*x + 9>,
    <"54663a", 1, [ -6, -1/8, 13/8, 1 ], [ 2, 4 ], [], "m", 
x^6 - 40*x^4 + 38*x^3 - 28*x^2 - 12*x + 9>,
    <"61320v", 1, [ 53/6, 1, 14/3, 1 ], [ 1, 4 ], [], "m", 
9*x^6 + 12*x^5 + 22*x^4 + 64*x^3 + 298*x^2 - 354*x + 93>,
    <"67032cf", 1, [ 2, 3/16, 5/4, 7 ], [ 2, 4 ], [], "m^*", 
21*x^6 + 126*x^5 + 210*x^4 + 63*x^3 + 21*x^2 - 63*x + 63>,
    <"67080r", 1/5, [ 7, 9, 37, 1 ], [ 2, 4 ], [], "g", 
x^6 + 6*x^5 + 8*x^4 - x^3 - 7*x^2 - 12*x>,
    <"67179c", 1, [ 6, 7/4, 31/4, 1 ], [ 2, 4 ], [], "m^*", 
49*x^6 + 294*x^5 + 455*x^4 + 42*x^3 - 91*x^2 - 252*x + 112>,
    <"70950bc", 1, [ 27/5, 1, 29/10, 1 ], [ 1, 4 ], [ 2 ], "g", 
100*x^6 - 60*x^5 + 125*x^4 - 70*x^3 + 185*x^2 - 120*x + 240>,
    <"72128j", 1, [ 8, 8/7, 31/7, 14 ], [ 2, 4 ], [], "m^*", 
56*x^5 + 98*x^4 - 14*x^2 - 112*x + 56>,
    <"75712bz", 17/7, [ 5, 4, 17, -26 ], [ 2, 2 ], [ 17 ], "", 
-104*x^6 + 104*x^5 - 234*x^4 - 676*x^3 + 2054*x^2 + 104*x - 1352>,
    <"76176cd", 1, [ 12, -1/4, 1, 1 ], [ 1, 4 ], [], "g", 
-x^6 - 12*x^5 - 24*x^4 + 82*x^3 - 84*x^2 + 60*x - 17>,
    <"76608bl", 11, [ 2/7, 7/6, -1/3, -14 ], [ 1, 4 ], [], "m", 
-27*x^6 + 90*x^5 - 534*x^4 + 96*x^3 - 1020*x^2 + 72*x - 504>,
    <"78400le", 1, [ 8, -4/7, -29/7, 1 ], [ 2, 4 ], [], "g^{(4)}", 
4*x^6 + 8*x^5 - 34*x^4 + 48*x^3 - 98*x^2 + 32*x + 12>,
    <"84474w", 49, [ 17, 3/4, 7, 7 ], [ 1, 4 ], [], "", 
84*x^6 + 756*x^5 + 2793*x^4 - 4158*x^3 + 17157*x^2 + 8568*x + 1008>,
    <"90650i", 1/2, [ -8, -1, 3, -35 ], [ 1, 4 ], [], "m^*", 
-35*x^6 - 280*x^5 - 280*x^4 + 840*x^3 + 1820*x^2 - 980*x + 140>,
    <"90950i", 1, [ 7, 8, 27, 1 ], [ 1, 4 ], [], "g", 
49*x^6 + 118*x^5 - 11*x^4 + 192*x^3 + 564*x^2 - 32*x + 64>,
    <"90950m", 1, [ 1, 4/5, 1, 5 ], [ 1, 4 ], [], "g", 
20*x^6 + 20*x^5 + 125*x^4 + 30*x^3 + 245*x^2 - 140*x + 100>,
    <"96300bd", 1, [ 8, 10/3, 43/3, 1 ], [ 1, 4 ], [], "g", 
-15*x^6 - 90*x^5 - 105*x^4 - 270*x^3 + 75*x^2 + 450*x + 225> ];

// The conductor of F = Jac C. The odd part was computed by Q. Liu's 
// program genus2reduction in Sage. The 2-part was computed in 
// September 2017 by T. Dokchitser and C. Doris.

conductors := [ 43850884, 90440100, 223143844, 231617961, 420332004, 
473976441, 191384375625, 532917225, 563777536, 596922624, 876006146304, 
684659556, 867008025, 1598720256, 184633744, 1676902500, 1731891456, 
523723225, 2951422929, 2988043569, 3760142400, 4493289024, 179989056, 
4513018041, 5033902500, 5202448384, 33808912384, 5802782976, 
710123065344, 6146560000, 17133191879076, 2054355625, 8271902500, 
8271902500, 9273690000 ];

// Generators for F(Q) modulo torsion where F = Jac C.

MWgens := [
    [
        [ x^2 + 3/2*x + 1/2, -6*x + 4, 2 ],
        [ x^2 + 5/2*x, -8*x + 2, 2 ],
        [ x^2 + x - 2, 4*x + 4, 2 ],
        [ x^2 + 3/2*x - 5/2, -4*x + 12, 2 ]
    ],
    [
        [ x^2 + 7/2*x + 3/2, 191/4*x + 93/4, 2 ],
        [ x^2 - 3*x + 2, -18*x - 14, 2 ],
        [ x^2 - 5/3*x - 2/3, 70/3*x + 10/3, 2 ],
        [ x^2 - 25/9*x - 16/9, -740/27*x - 380/27, 2 ]
    ],
    [
        [ x + 1, 2*x^3 + 16, 2 ],
        [ x^2 - 5/2*x, 24*x - 4, 2 ],
        [ x, -2*x^3 + 4, 2 ],
        [ x - 1/6, 2*x^3 + 17/12, 2 ]
    ],
    [
        [ x^2 - 9/16*x + 5/16, 891/256*x + 1377/256, 2 ],
        [ x^2 - 9/4*x + 35/4, 243/16*x + 243/16, 2 ],
        [ x^2 - 2*x + 3, -10*x - 13, 2 ],
        [ x^2 - 45/29*x + 31/29, 4536/841*x + 6480/841, 2 ]
    ],
    [
        [ x, -6*x^3 + 12, 2 ],
        [ x, 6*x^3 + 12, 2 ],
        [ x + 1, -6*x^3 - 24, 2 ],
        [ x + 1/3, -6*x^3 - 70/9, 2 ]
    ],
    [
        [ x^2 - 33/28*x - 4/7, -5967/392*x - 270/49, 2 ],
        [ x^2 - 3/2*x - 1, -9*x, 2 ],
        [ x^2 - 10/3*x - 4/3, 181/9*x + 64/9, 2 ],
        [ x^2 - 51/10*x - 8/5, 1647/25*x + 432/25, 2 ]
    ],
    [
        [ x^2 + 135/8*x - 9/8, 2905/64*x - 735/64, 2 ],
        [ x^2 + 16*x - 121/6, -119/2*x + 329/6, 2 ]
    ],
    [
        [ x - 1/3, 3*x^3 - 14, 2 ],
        [ x^2 - 19/3*x - 14/3, -115/3*x - 50/3, 2 ],
        [ x^2 - 22/3*x + 7/3, 122/3*x + 1/3, 2 ],
        [ x^2 + 2*x + 1/3, 40*x + 5, 2 ]
    ],
    [
        [ x - 3, -2*x^3 - 2, 2 ],
        [ x^2 - x - 6, -8*x - 32, 2 ],
        [ x^2 - x - 2/3, -8/3*x, 2 ],
        [ x + 1/2, 2*x^3 - 3/2, 2 ]
    ],
    [
        [ x^2 - 11*x + 7, 150*x - 100, 2 ],
        [ x^2 - 13/3*x + 1/3, 550/9*x - 100/9, 2 ],
        [ x^2 - x + 1/3, -100/3*x + 50/3, 2 ],
        [ x^2 - 43/3*x + 1/3, 700/9*x + 50/9, 2 ]
    ],
    [
        [ x^2 - 1/2*x - 26/3, 221/6*x + 578/3, 2 ],
        [ x - 2, -4*x^3 - 70, 2 ],
        [ x + 1/4, 4*x^3 + 223/16, 2 ],
        [ x^2 + 25/4*x + 3/2, 103/2*x - 1, 2 ],
        [ x^2 + 78/11*x + 72/11, -8433/121*x - 13088/121, 2 ],
        [ x^2 + 129/14*x + 166/7, 255/98*x - 3978/49, 2 ]
    ],
    [
        [ x^2 - 5*x + 4, 22*x - 46, 2 ],
        [ x^2 - 3*x - 4, 12*x - 6, 2 ],
        [ x^2 - 9/5*x - 2/5, 66/5*x - 12/5, 2 ],
        [ x^2 - 69/13*x + 2, -9198/169*x + 504/13, 2 ]
    ],
    [
        [ x + 2, 3*x^3 - 15, 2 ],
        [ x - 4, 3*x^3 - 57, 2 ],
        [ x^2 + 3*x + 2, 29*x + 19, 2 ],
        [ x^2 - x + 1/2, -3/2*x + 7/2, 2 ]
    ],
    [
        [ x^2 + x, 10*x - 2, 2 ],
        [ x, -2*x^3 - 2, 2 ],
        [ x^2 - 1/3*x + 4/3, -130/9*x + 82/9, 2 ],
        [ x + 4, 2*x^3 + 2, 2 ]
    ],
    [
        [ x^2 + 3/2*x + 1, 11/4*x - 1/2, 2 ],
        [ x^2 + x - 5/2, -9/2*x + 8, 2 ],
        [ x^2 - 4*x + 2, 21*x - 11, 2 ],
        [ x - 2/3, x^3 - 11/9, 2 ]
    ],
    [
        [ x^2 + 3*x, -30*x - 10, 2 ],
        [ x^2 + 1/2*x - 10, 45*x + 90, 2 ],
        [ x^2 + 1/2*x, -15*x + 10, 2 ],
        [ x^2 + 11*x + 24, 170*x + 590, 2 ]
    ],
    [
        [ x - 1, 3*x^3 - 1, 2 ],
        [ x^2 - 1, 7*x - 5, 2 ],
        [ x + 5, -3*x^3 - 103, 2 ],
        [ x^2 + 3/5*x - 8/5, 122/25*x - 172/25, 2 ]
    ],
    [
        [ x, x^3 - 2, 2 ],
        [ x^2 - x + 3, -6*x - 1, 2 ],
        [ x, x^3 + 2, 2 ],
        [ x + 1/6, x^3 - 19/36, 2 ]
    ],
    [
        [ x, x^3 - 3, 2 ],
        [ x, x^3 + 3, 2 ],
        [ x - 3, -x^3 - 51, 2 ],
        [ x^2 + x - 3, 4*x - 24, 2 ]
    ],
    [
        [ x, -x^3 - 3, 2 ],
        [ x, x^3 - 3, 2 ],
        [ x + 1/2, x^3 + 1, 2 ],
        [ x^2 - 7*x - 2, -29*x - 11, 2 ]
    ],
    [
        [ x^2 + 7/4*x - 1/2, -355/16*x + 5/8, 2 ],
        [ x - 1/4, 3*x^3 + 39/8, 2 ],
        [ x^2 + 5/6*x + 1/6, -203/12*x + 119/12, 2 ],
        [ x - 1, 3*x^3 + 9, 2 ]
    ],
    [
        [ x^2 + 1/2*x - 3/2, 63/4*x + 21/4, 2 ],
        [ x^2 + 4*x + 9, 21*x - 42, 2 ],
        [ x^2 + x + 3/2, -21/2*x - 21/2, 2 ],
        [ x^2 - 3/2*x + 1/2, -105/4*x + 21/4, 2 ]
    ],
    [
        [ x, x^3, 2 ],
        [ x^2 + x, -3*x, 2 ],
        [ x^2 + 2*x, x, 2 ],
        [ x - 2, -x^3 - 10, 2 ]
    ],
    [
        [ x + 1, -7*x^3 - 28, 2 ],
        [ x - 1/2, -7*x^3 - 7/4, 2 ],
        [ x + 1, 7*x^3 - 14, 2 ],
        [ x^2 + 4*x + 3, -14*x - 35, 2 ]
    ],
    [
        [ x^2 + 3*x + 2, -80*x - 50, 2 ],
        [ x - 1/2, -10*x^3 + 65/4, 2 ],
        [ x^2 + 3/2, -10*x + 15/2, 2 ],
        [ x^2 + 3/2*x - 1, 50*x - 10, 2 ]
    ],
    [
        [ x + 1, 14, 1 ],
        [ x^2 - x - 2, -14*x - 28, 2 ],
        [ x^2 + 3/4*x - 1/4, -7*x + 7, 2 ],
        [ x^2 - x - 1/4, 35/2*x - 21/4, 2 ]
    ],
    [
        [ x^2 + 1/2*x - 11/6, 52/3*x - 52/3, 2 ],
        [ x^2 - 5/6*x + 13/6, 416/9*x + 104/9, 2 ]
    ],
    [
        [ x^2 - x + 1/2, 9/2*x, 2 ],
        [ x^2 - x + 7/5, -36/5*x + 18/5, 2 ],
        [ x^2 - x + 23, 36*x - 126, 2 ],
        [ x^2 - 43/25*x + 17/25, -1818/625*x + 2142/625, 2 ]
    ],
    [
        [ x^2 - 2/3*x + 2/3, -16*x + 16, 2 ],
        [ x^2 - 2/3*x + 6, 32*x + 48, 2 ],
        [ x^2 + 10/21*x + 38/21, 608/49*x - 176/49, 2 ],
        [ x^2 + 10/291*x + 302/291, 52600/9409*x + 59024/9409, 2 ]
    ],
    [
        [ x^2 - x + 2, -10, 2 ],
        [ x - 1/3, -2*x^3 - 32/9, 2 ],
        [ x^2 - 7/3*x + 2/3, 52/9*x - 50/9, 2 ],
        [ x - 2, -2*x^3 + 22, 2 ]
    ],
    [
        [ x^2 + 4*x + 3, 231*x + 105, 2 ],
        [ x^2 + 4*x + 3, -357*x - 483, 2 ],
        [ x^2 + 5*x + 1, -258*x - 57, 2 ],
        [ x^2 + 3544/697*x + 1272/697, -131405547/485809*x - 
        62672988/485809, 2 ]
    ],
    [
        [ x^2 + x - 2, 35*x, 2 ],
        [ x^2 - 4/3*x + 1/3, 385/9*x - 70/9, 2 ],
        [ x^2 + 17/3*x - 13/3, 70/9*x - 245/9, 2 ],
        [ x^2 - 2*x + 1, -35*x, 2 ]
    ],
    [
        [ x^2 + 31/11*x + 16/11, -1666/121*x - 3128/121, 2 ],
        [ x^2 + 53/72*x + 1/6, -80665/5184*x + 2635/432, 2 ],
        [ x^2 + 3/2*x - 3/2, -35/4*x - 55/4, 2 ],
        [ x^2 + 4*x, -70*x - 8, 2 ]
    ],
    [
        [ x^2 + 2*x - 1, 35*x - 5, 2 ],
        [ x^2 + 3, 5*x - 5, 2 ],
        [ x^2 + 1/2, -10*x + 15/2, 2 ],
        [ x^2 - 1/2*x + 3/2, 10*x - 10, 2 ]
    ],
    [
        [ x^2 + 2*x, 15*x - 15, 2 ],
        [ x^2 + 5*x + 15/4, -195/4*x - 105/2, 2 ],
        [ x^2 + 3/34*x - 20/17, -14685/1156*x - 435/289, 2 ],
        [ x^2 + 2*x, 30*x + 15, 2 ]
    ]
];

// Component group data at the bad primes for C. The entries listed are :
// Cremona label for E, bad primes for C, 
// component group orders (Liu/Sage), component group orders (Donnelly/Magma).  
// Cases where an answer was not returned are indicated by 0.

cgdata := [
    <"6622b", [ 2, 7, 11, 43 ], [ 0, 2, 7, 1 ], [ 28, 2, 7, 1 ]>,
    <"9510e", [ 2, 3, 5, 317 ], [ 0, 4, 17, 1 ], [ 0, 4, 17, 1 ]>,
    <"14938n", [ 2, 7, 11, 97 ], [ 0, 2, 2, 1 ], [ 146, 2, 2, 1 ]>,
    <"15219c", [ 2, 3, 19, 89 ], [ 0, 16, 1, 1 ], [ 1, 16, 1, 1 ]>,
    <"20502ba", [ 2, 3, 17, 67 ], [ 0, 16, 2, 1 ], [ 146, 16, 2, 1 ]>,
    <"21771b", [ 2, 3, 41, 59 ], [ 0, 16, 1, 1 ], [ 1, 16, 1, 1 ]>,
    <"23025c", [ 2, 3, 5, 19, 307 ], [ 0, 62, 16, 7, 2 ], [ 1, 62, 16, 7, 2 ]>,
    <"23085h", [ 2, 3, 5, 19 ], [ 0, 9, 31, 2 ], [ 1, 9, 31, 2 ]>,
    <"23744v", [ 2, 7, 53 ], [ 0, 2, 1 ], [ 16, 2, 1 ]>,
    <"24432j", [ 2, 3, 5, 509 ], [ 0, 1, 1, 1 ], [ 16, 1, 0, 1 ]>,
    <"25296a", [ 2, 3, 17, 31, 37 ], [ 0, 2, 4, 7, 7 ], [ 16, 2, 4, 7, 7 ]>,
    <"26166h", [ 2, 3, 7, 89 ], [ 0, 49, 9, 1 ], [ 0, 49, 9, 1 ]>,
    <"29445c", [ 2, 3, 5, 13, 151 ], [ 0, 31, 2, 2, 1 ], [ 1, 31, 2, 2, 1 ]>,
    <"39984bf", [ 2, 3, 7, 17 ], [ 0, 8, 9, 1 ], [ 16, 8, 9, 1 ]>,
    <"40764a", [ 2, 43, 79 ], [ 0, 1, 2 ], [ 9, 1, 2 ]>,
    <"40950u", [ 2, 3, 5, 7, 13 ], [ 0, 16, 16, 2, 1 ], [ 56, 16, 16, 2, 1 ]>,
    <"41616l", [ 2, 3, 17 ], [ 0, 4, 9 ], [ 16, 4, 9 ]>,
    <"45770c", [ 2, 5, 23, 199 ], [ 0, 7, 2, 1 ], [ 1, 7, 2, 1 ]>,
    <"54327c", [ 2, 3, 7, 13, 199 ], [ 0, 2, 2, 2, 1 ], [ 1, 2, 2, 2, 1 ]>,
    <"54663a", [ 2, 3, 7, 19, 137 ], [ 0, 2, 2, 2, 1 ], [ 1, 2, 2, 2, 1 ]>,
    <"61320v", [ 2, 3, 5, 7, 73 ], [ 0, 31, 4, 17, 1 ], [ 8, 31, 4, 17, 1 ]>,
    <"67032cf", [ 2, 3, 7, 19 ], [ 0, 16, 16, 2 ], [ 4, 16, 16, 2 ]>,
    <"67080r", [ 2, 3, 13, 43 ], [ 0, 4, 1, 2 ], [ 8, 4, 1, 2 ]>,
    <"67179c", [ 2, 3, 7, 457 ], [ 0, 2, 16, 2 ], [ 1, 2, 16, 2 ]>,
    <"70950bc", [ 2, 3, 5, 11, 43 ], [ 0, 2, 16, 2, 1 ], [ 119, 2, 16, 2, 1 ]>,
    <"72128j", [ 2, 7, 23 ], [ 0, 16, 2 ], [ 16, 16, 2 ]>,
    <"75712bz", [ 2, 13, 17 ], [ 0, 16, 7 ], [ 16, 16, 7 ]>,
    <"76176cd", [ 2, 3, 23 ], [ 0, 16, 1 ], [ 16, 16, 1 ]>,
    <"76608bl", [ 2, 3, 7, 11, 19 ], [ 0, 16, 1, 7, 1 ], [ 0, 16, 1, 7, 1 ]>,
    <"78400le", [ 2, 5, 7 ], [ 0, 4, 4 ], [ 16, 4, 4 ]>,
    <"84474w", [ 2, 3, 7, 13, 19 ], [ 0, 16, 2, 2, 1 ], [ 46, 16, 2, 2, 1 ]>,
    <"90650i", [ 2, 5, 7, 37 ], [ 0, 16, 16, 2 ], [ 1, 16, 16, 2 ]>,
    <"90950i", [ 2, 5, 17, 107 ], [ 0, 4, 8, 1 ], [ 4, 4, 8, 1 ]>,
    <"90950m", [ 2, 5, 17, 107 ], [ 0, 16, 2, 1 ], [ 46, 16, 2, 1 ]>,
    <"96300bd", [ 2, 3, 5, 107 ], [ 0, 16, 9, 2 ], [ 9, 16, 9, 2 ]>
];

// Some data allowing us to quickly recover the points we computed on 
// the Kummer surface. These are used to verify the congruence.

kumsrf_pts := [
< "6622b1",20*x^6 + 44*x^5 - 23*x^4 - 10*x^3 + 81*x^2 - 52*x + 4,
[ 14, -1, 20, 17, -21, -13, -3, -5, -4, -3, 3, 4, 14, 8, 12, -27, -410, -13, 2, 0, -12, 
-16, -24, -4, -136, -8, -4, -16, 4, 8, -2, 12, 362, -4, -10, -8, 6, 16, 0, 20, -184, 0, 
10, 8, -16, -22, -3, 3, -17176, 138, 620, 247, -55, -433, -228, 328, -12371, -679, -408, 
349, -63, -700, -194, 40 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 43, 8, -3, 5, -6, -7, -3, 3 ],
[ -23502, -308, 169, 1343, -628, -1473, -209, -204 ],
[ -2026670, -12058, 104975, 48927, -88462, -112149, -59955, 76512 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -92, 2, 4, -9, 6, 8, -3, 3 ],
[ -34876, -794, 806, 1047, 1649, 588, 1070, -2282 ],
[ 5284018, 260544, 4527, 37070, -102104, 10453, -15028, 11083 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -12, 5, 3, -5, 3, 6, 0, 4 ],
[ -5100, -631, 571, -315, 172, 539, 153, -43 ],
[ -292077, 3503, 25724, -33750, 3531, 20549, -2126, 13706 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -12, 332, 312, -64, 60, 92, -164, 124 ],
[ 8404224, -1954800, -1607856, -761120, 909040, 2897968, 1910784, 239440 ],
[ -255320861824, 7225553600, 15131836736, 14797683200, -5876291328, 2363052352, 
7267858304, -6454194304 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 454, 1268, 494, 38, 652, -82, -270, -160 ],
[ 192158150, -11011870, -684253, -2934577, -13030850, -3760541, -1509043, -2541584 ],
[ -633774976590, 484722944814/7, 138417936501/7, 33305120809/7, 348557921458/7, 1882382819, -86898235749/7, 4052719600 ]
]
] >,

< "9510e1",9*x^6 - 12*x^5 + 2*x^4 - 84*x^3 + 437*x^2 + 528*x + 144,
[ 692, 26, -14, 19, -16, 0, 3, 0, 1008, -5, -27, 4, -3, -1, 13, -1, 416, 20, 2, 10, -21, 
22, -9, 13, 221, 12, -21, -4, -5, -8, 6, 5, 647, -14, -13, 3, -20, 2, -5, 0, 264, 10, -7, 
-14, 2, -3, -11, -11, 16252, 1286, -212, 264, -160, -546, 69, -75, -29757, 339, 1913, 
-172, -331, 281, -484, -236 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -340, 0, 5, 2, -3, 0, -3, 0 ],
[ 86782, -685, 64, -3486, 1473, -351, 925, -64 ],
[ -13379710, 363127, -8791, 270726, -345831, 296337, 8177, -200807 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -205, 31, -14, 35, -21, 25, 0, -3 ],
[ -23457, -15181, 20874, -641, 17693, -12435, -736, 1746 ],
[ 156657108, -4005802, -7876475, 3210334, -1456951, -1721693, 3381718, 5987244 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -97, -13, 1, -23, 6, -11, 3, 1 ],
[ 186618, -2466, 8420, 5852, 76, 2810, -1502, 1904 ],
[ -63071692, 2853232, 378836, -460524, 1887564, -145428, -719548, -256100 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -2214, 1362, -266, -650, -364, -194, -94, -218 ],
[ 240829320, 3809400, 2977104, 18930960, 12613680, -13236120, 12741096, 10292352 ],
[ 14215479293024, -223227104768, -791205399712, -29778624288, -382814356320, 197838956832, -107103041824, 75843660448 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 1820, -12962, -9340, -22158, -32058, -4620, 98, 8260 ],
[ -140749803712, 18361793611, 11009760650, 190289853, -12043041777, 15052192674, -18051161899, -6331218902 ],
[ -965785532449473968/7, -1738819182532723, 39854707659815530/7, -1253150218560339/7, 12632985366889311/7, -11180054832733950/7, 5607557061644869/7, 6313801034986250/7 ]
]
] >,

< "14938n1",4*x^6 + 36*x^5 + 37*x^4 - 150*x^3 - 47*x^2 - 72*x + 16,
[ 603, 0, 37, 19, -25, 21, -7, -14, 570, -2, -31, 2, 26, -19, 32, -1, -1754, 20, 33, 6, 
-18, 38, 16, -18, -824, -26, 11, 25, 7, -15, -9, -19, 1422, 16, -8, -14, 16, 7, -21, 10, 
879, 15, -14, -5, -4, 8, 23, -2, -9383, 986, 838, 1056, -977, -964, 736, -351, 3126, 3158,
841, -1587, 1035, 1008, -1299, 155 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -56, -4, -4, -4, 6, -2, -2, 2 ],
[ 25043, 1266, 198, 788, -1207, 386, 720, -303 ],
[ 747477, -8770, -92282, -11196, 89547, -1746, -9860, 65615 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 601, 13, -3, 1, -7, 12, 9, 4 ],
[ -404190, 5815, -4506, 1438, -1261, 11488, 11119, -2756 ],
[ -146166584, 2398699, -487082, 1237181, 308640, 4344212, 1927597, 373401 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 273, -22, -5, -1, 3, 7, -8, -5 ],
[ 178528, -7490, 1334, 3080, -2148, 4416, -5748, -4274 ],
[ 2804415, 275034, -269294, -74412, 70025, 149154, -426412, -131187 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -472, -432, -4, -524, 344, 108, -80, -200 ],
[ 263136848, -3068544, 5464192, 8204608, -5162064, -2572704, 510080, 1519312 ],
[ 2455706881344, -92286368896, 30978638208, -49796591360, -68034778944, 74422144, 55796258048, -77333044288 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -25, -247, -71, -178, 191, -58, -107, 2 ],
[ 41836938, 920423, 494144, 657419, -1596035, -41986, 364959, -752157 ],
[ -72416303137, -18952912624/7, -7903565643/7, -16256647795/7, 22174249057/7, -1758552539/7, -7020572771/7, 6209166084/7 ]
]
] >,

< "15219c1",-7*x^6 + 18*x^5 - 93*x^4 + 18*x^3 + 117*x^2 - 36*x + 64,
[ 46, -29, 21, 5, 12, -22, 9, -1, 168, -4, 10, 32, 9, 18, 5, 5, -657, -5, 27, 30, 29, 23, 
-13, -6, -159, -12, 24, 14, -5, -2, 9, -3, 328, 5, -4, -31, -18, -20, -6, 5, 838, 3, -21, 
-15, -5, 6, -11, 1, 4595, 423, -236, -1091, -1801, -54, 540, -140, 14988, 201, 39, -251, 
-124, -123, 440, 336 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 52, -27, 25, -2, -5, 0, 4, -3 ],
[ -302433, -13059, 16696, -6561, -4500, -6196, -4530, -443 ],
[ 15773582, 127968, 637255, -163338, -725500, 221134, 383945, -232629 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 915, 57, -18, -75, -99, -42, 36, -21 ],
[ 1525014, 20034, -55350, -46548, -18279, 3024, 17172, -23868 ],
[ -94667967, -3439233, -10828431, -39575817, -20489733, -7479270, 17838063, -10565181 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 683, 69, -31, 53, -46, -3, 17, -3 ],
[ -1578972, 50565, 24958, 43455, -67524, 14999, 37410, -17672 ],
[ -293708630, 15297201, -9775687, -5462076, -16439567, -5736571, 12888283, 2021964 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 1486, -1146, -1226, -2654, -3344, -354, -914, 378 ],
[ 4601750760, 6523044, -88805552, -183640620, -239799504, 48981188, 187604952, -140670032 ],
[ 266644782041456, -729083292360, 13403649411448, 852372406704, -8833760082376, -2932978141592, -5220497633656, -1183842141360 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 1380, -111, -1983, -2667, -879, -2361, -3231, -858 ],
[ 6885932625, 62141244, 192583167, 195348066, 45862365, 208048398, 278704269, 40583472 ],
[ -1020732185243809/7, -7434725986389/7, -36581923342659/7, -42060882939546/7, 
-1702802025537, -41190050707455/7, -55283772372453/7, -1596816909588 ]
]
] >,

< "20502ba1",36*x^6 + 108*x^5 + 381*x^4 + 474*x^3 + 849*x^2 + 504*x + 144,
[ 178, -6, -11, 1, -6, 3, -1, 2, 14, 8, 5, -11, -5, -7, -1, -1, -258, 37, 10, -16, -3, -7,
22, -13, 44, -5, 3, 2, 6, 2, -3, 5, -92, 17, 7, 3, 9, 7, 1, 2, -107, 6, -16, -14, -1, -3, 
8, 6, 256, 384, -10, -114, -65, 38, 78, -110, -60, -127, 31, -133, -73, -107, -58, -55 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -17, 0, 2, -1, 0, 1, 1, 1 ],
[ -4213, 50, 559, 288, 166, 333, -144, 47 ],
[ 249060, 17781, -4559, -6167, -5964, -21676, 6641, 145 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 72, 8, -6, -18, 3, 1, 7, 4 ],
[ 16455, -1040, 2957, 1119, -313, 918, 1006, -1644 ],
[ -675675, 63947, -29105, -216133, -1405, -3953, 59503, 21839 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -108, -15, -1, 9, -6, 4, -5, 1 ],
[ 33692, -858, 158, 262, 188, -60, 198, 262 ],
[ 1102078, -178737, -55001, 25595, -14330, 4430, -71981, 23851 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -432, 2712, -1056, -2880, -2112, -3504, 1320, 528 ],
[ 1763533728, 33138864, -40542480, -2366928, -8811360, -121075488, 47274480, 41943600 ],
[ 12542566518144, 3458213542080, -2110307943744, -1027926573120, -434359103616, -1856848109184, 1471572273600, 3203651520 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -1095, -8508, 10929, 5355, 4560, 12621, -6783, -726 ],
[ 22210543947, 1318938573, -1657113591, -820040499, -707519130, -1939671012, 1032041913, 106505421 ],
[ -5915501659691129/7, -465154250714490/7, 590399800474287/7, 41547760490871, 249424173854736/7, 686788064048307/7, -52444468452507, -5504965003464 ]
]
] >,

< "21771b1",4*x^6 + 36*x^5 - 87*x^4 - 468*x^3 + 657*x^2 + 648*x + 128,
[ -333, 11, 22, -5, -7, -2, 15, 11, 305, 57, 5, -16, -6, 4, 6, 6, 362, -12, -9, 34, -15, 
-2, -24, 21, 492, -45, -21, 31, 2, 0, -24, 1, -56, -3, -4, 27, 1, -7, -5, 16, 223, -9, 
-11, 19, -2, -3, -13, 12, 7912, -1654, 35, 847, 14, -14, -315, -14, 32613, -1926, -803, 
1299, 346, 64, -1304, -101 ],
[
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -444, 50, 34, -22, 18, 14, 14, 0 ],
[ -194348, -18880, -16908, 18908, -2344, 1228, 980, 2512 ],
[ -22164576, -69056, -672152, 423984, -12832, -1309128, 169320, 514920 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -504, 69, 54, -39, 6, 30, 54, 18 ],
[ 208638, -52281, 10881, -13248, -58626, -20484, -41391, -18810 ],
[ 326727459, 60292593, 30357855, -57355452, -27425466, 28515159, -2784834, -6712875 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 648, -20, -22, 4, 6, -20, -20, -6 ],
[ 187732, -15832, -16944, 8372, -6424, -14468, -18700, -6836 ],
[ 26867856, -17832640, -7684408, 9203760, -3840632, -4661304, -8942856, -3212616 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 4056, 9472, 5648, -4376, 4224, 1048, 1984, 600 ],
[ -8028059840, -56577664, 61682880, 78891776, -117977920, 278232832, 327881600, 7030912 ],
[ -282516555635712, -20154469193728, -22146524790784, 24847392050688, 5424661212160, -2758658351616, -3975482161152, 3595439947776 ]
],
[
[ 1, 0, 0, 0, 0, 0, 0, 0 ],
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