curve #1870

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y² = x³ + x² − 460436012490601812655247863486153221460x + 3817078696600062059259879370390852023073837100587251924608

a-invariants

[0, 1, 0, -460436012490601812655247863486153221460, 3817078696600062059259879370390852023073837100587251924608]

rank (lower bound)

≥ 24

torsion subgroup

trivial

conductor (N)

82169019903608474432382828776526797358638326332455264364993047784023758183619634843480

discriminant (Δ)

-47044021044657740057212169610273266175854358159659297862094167560127764447406563300477394889806195741279157116000000

Faltings height

21.0487

naive height

278.6968

primes of bad reduction

2, 3, 5, 7, 13, 19, 23, 29, 37, 53, 67, 83, 13700399, 511842055478901105598431188597, 7764372498274348111061126615225581

regulator

341291769928754887975767.97920441594451352359376447277944342431265012091671509707190655

submitted by

Ritik Jain

submitted at

2026-09-25 17:21:10 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 17:26:02 UTC

Witness: 24 independent points log in to add more points →

(3171919171834338169447137606109413946768749077024/72619650503726499904888427209, -5066832462562382571068491065042252076973368157064952462891272708169212000/19569567684965464585097823601461459653597573)
(-12237613568382973171/4, 576727448140715465190817801035/8)
(-103080332743956054918134/6561, -45009433195407860650128931618225790/531441)
(-18497689330799406214, 77490826525803501344504675250)
(-923666770030191210967526/38809, -274997862276832939541293194925121850/7645373)
(22996852173994748180086, 3487405124020498513438223678624850)
(1300022397136298041598726/5041, 1477292006765800838472900396197231250/357911)
(-382407081930101699359/16, -2188225132989727856179041685305/64)
(978939507580696615306/121, -33182191299395880255284956171350/1331)
(-12477212274424403928346/529, 479871522150968488634155861008210/12167)
(11849134229405028836, 4995248860563141566197796400)
(17239239355862455510586, -2263480790662712479983068840420850)
(1215414672945439847834/49, 30014607762874670472630129424830/343)
(16700314703958254836, 28024608744708196003541100600)
(18694277887895520868496/961, -1404154192074063690781299272200200/29791)
(63288709121843099000126/6241, 6809068302803562192318405769453350/493039)
(313033406583448015188652736411546/154289165699761, 103049753843580570703753680007846882469564235013950/1916474945400589604759)
(-9550280359829234629, 85693121765070553115061563520)
(12594073254724519136, 3983511454317636531013105200)
(372675894324445922425876/32041, 34017388969008578403296896011706200/5735339)
(12127735101757903136, 4099236410158928386456903200)
(-83572358890653964, 62093138401794513338370045000)
(16233195678960197069/4, 359139509914270117411613795925/8)
(11941265650663254302, 4651957098193990752676822950)

Commentary

Specialization at T = 117/100 of the rank-17 fibration of X1092 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 246846T^2 + 383054T + 42417, p = 514599T^2 + 1028123T + 13068, q = 328107T^2 - 253125T - 3900, D = 122691873330T^4 + 5928867377T^3 - 269578152744T^2 + 86183018989T + 23435297028, E = 83083896090T^4 + 197291268661T^3 - 228106232536T^2 + 14398475825T + 2317084900.

last edited by wgxli at 2026-09-25 18:15:01 UTC · history