curve #815

y² + xy = x³ − 4704485889547869615054230388002457799527150x + 3911543169657028376580361726357230437854526531905552305617722500

a-invariants

[1, 0, 0, -4704485889547869615054230388002457799527150, 3911543169657028376580361726357230437854526531905552305617722500]

rank (lower bound)

≥ 26

torsion subgroup

trivial

conductor (N)

2090052314595515427034152500520120902146273491021421311412900199956586310420257693417770027719997020373470

discriminant (Δ)

54042652364004468697205306256620589849594814325453900021062402371369358566895451129178861873106225446892580291379510421382400000

Faltings height

23.3594

naive height

306.3849

primes of bad reduction

2, 3, 5, 11, 13, 29, 41, 43, 61, 79, 3710880413, 16759958116770898951, 2445678624550988535007813011149135039712141342841512502572393469

regulator

20329616818415461343770279453.7116026872027643499553889734867050022518663014845497727820865

submitted by

thomas

submitted at

2026-09-21 19:30:58 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-21 20:05:16 UTC

Witness: 26 independent points log in to add more points →

(1374405928575079984650, -6473886305704299882165978572250)
(285100477704581221851710025/215296, -196505154375150972820167438825242563725/99897344)
(1697207465567026473060, -28563666992650358392496391756510)
(463087946741659460684265/1024, -1419464328937510340519067705157427205/32768)
(3595961560127971985952, -183012442966079427408568328322654)
(1111083500659399061520, 7490418731873454476549968297230)
(2163347995221307665750, 62118700434333145910853723512100)
(-289501502222203901370048/1681, -4733342977822564357416452078613061074/68921)
(22054460393686865810445/16, -433242900141715870434540136302405/64)
(11500955092897597900485/16, -1921400027105828995187658094367865/64)
(250900464007728681895946005/242064, 1453170030565382706430764094095680311915/119095488)
(506020639655186872656, -40749800990075655246026014161438)
(-2330778367612108720800, 47059949390826716819308612236450)
(271695798947052579066561/256, -43744855286304354562208683312642503/4096)
(385295763143396483126100, 239157763472615911741480232895694050)
(1185541792515558103692, 684543478245516207441660533106)
(1331744544926853956250, 2876921053635189320207544729600)
(36226287451599798972317844/49, 218038846803561828476321130469937536806/343)
(113361192416691858972762, -38160818601961210842002391141958848)
(42878287856266588530156, 8867692099098430042892092207442562)
(2644912986870085509667791900/3157729, 132633559681912202276885403046683321343650/5611284433)
(1051816515939455988357960060/10201, 34104732241916686009317680498777684943450/1030301)
(4324298872138091765138961620/4190209, -106999694921945278740013677808166806562890/8577357823)
(1765671721608869237059848600/2209, 74193070532324067185708545376338243712250/103823)
(9905357890663548290472, -963943546114056473677870346643774)
(1828089592186501751900, -37691314834129927535175829441150)

Commentary

specialization at T=355/767 of rank-17 fibration of #302 family: y^2 - (Lx+B)y = x(x+D)(x+E), where L = 446667T^2 + 471466T + 239031, p = 318552T^2 + 368554T - 72570, q = 733413T^2 - 45082T - 14960, D = 5(7174492962T^4 - 7114589515T^3 - 22069002960T^2 + 3909144679T - 205134150), E = 882769396002T^4 + 811447034567T^3 - 1174040743T^2 - 32493137198T - 2386325360

last edited by thomas at 2026-09-21 19:38:37 UTC · history