curve #4108

y² + xy = x³ − 9882697330631632842885969936728116820209114723771x + 39846882748632928639451929038284388824128635346592921012307735996626765201

a-invariants: [1, 0, 0, -9882697330631632842885969936728116820209114723771, 39846882748632928639451929038284388824128635346592921012307735996626765201]
c-invariants: [474369471870318376458526556962949607370037506741009, -34427706694818850344486467400631919749524705627246119199058294956141785245177]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 1213249524563722385101574358389528895083959754877740265336233395688563597023994735590
discriminant (Δ): -624144291499869793483633050099603743097969360900672930318672052507220688727755817933474305771393064963270595742330237843386118254198599425159639859200
Faltings height: 27.4269
naive height: 352.4655
primes of bad reduction: 2, 3, 5, 7, 13, 17, 29, 59, 67, 71, 73, 79, 107, 137, 163, 593, 659, 929, 4903, 8821, 9431, 10369, 55343, 381739, 414737, 565891, 678553, 1639459
regulator: 18213594436607546135466.6253825752304024232600212584432159860491792143809722189
submitted by: wgxli
submitted at: 2026-10-09 15:23:15 UTC
last updated: 2026-10-09 15:23:15 UTC

Witness: 20 independent points log in to add more points →

(165935700459318597125738, 6181549921111238304016076841685310825)
(215533694663635050567116, 6142217925854875040909760537722828147)
(399853240901573621958446, 5996597683153641080742723539992010417)
(733799521573524602912366, 5743699778120694481677339655888526897)
(-4341808793336093657046910, 952317800675157393380924821633388549)
(1096145415349257048056126, 5507364724587712657288048431072539057)
(1578104762616948300243614, 5308586085838631985183145380201059153)
(1692318062886641433843614, 5288564423343901836898932839142179153)
(2062936995714978359721668, 5314015472928525832391987648570162897)
(3051568258953636181531526, 6172970999126687867036286891520184057)
(3159219269799925831192526, 6336906856154943935902817669422868417)
(3770321235677716831080776, 7495483331712435287351730753337667807)
(4262674774637132745672158, 8670331470316988105345759554986135377)
(-3938478320568970275098386, 4204465054766072102209903869736424753)
(-4243249071726963228866584, 2319749808666434220613809873172319897)
(7869124573167968496651230, 21198100475177675074524476686439130449)
(8038706514295944406489766, 21905945419784242626511485510699512897)
(9995265922069721061619046, 30653664442623639399950414380994950337)
(19840329713549776039072478, 87485382612182931260023674061632362705)
(-16971986195852715505702505506/6889, 4012348382189439797629950121107281778241819/571787)

Commentary

specialization at T = 39757/56373 of the rank-9 fibration of X720 given by: y^2 + (355T - 3074)(T + 34)xy = x(x + (181 - 7T)(32T + 259)(47T + 529)(T - 97))(x + (12T - 371)(83T - 47)(6T + 23)(42T + 47)).

last edited by wgxli at 2026-10-09 15:23:15 UTC · history