curve #4079

y² + xy + y = x³ − x² − 237843590108509841765428899880454035832063133466292x + 1187109867088825244978128704282411400339941147142161032935981794570643124759

a-invariants: [1, -1, 1, -237843590108509841765428899880454035832063133466292, 1187109867088825244978128704282411400339941147142161032935981794570643124759]
c-invariants: [11416492325208472404740587194261793719939030406382001, -1025662925164745011661103149125787986455583329798184758278616530783398831073001]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 59922190121336398795572364641929095928611712492963586620439649207478009865597394590
discriminant (Δ): 252314177355834588647381366058696061681393231871640909455867016287591582767500657939435797429942358280353320453798376954890322537767229523376640000000000
Faltings height: 27.9639
naive height: 359.6007
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 79, 89, 103, 151, 167, 229, 263, 283, 307, 313, 479, 2381, 2621, 3919, 5431, 10459, 12101, 13441, 14851, 16481, 35509, 260111
regulator: 691585808662145698497.025991104914093588882682474311012986273756590247057429169
submitted by: wgxli
submitted at: 2026-10-09 05:31:20 UTC
last updated: 2026-10-09 05:31:20 UTC

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

(47067844897071791209417, 34291619765667107220961491959410673541)
(5814412815903811632520917, 871325108938918635965997094824804541)
(5816962178859076360268517, 641508506725873816768148739320214541)
(5681684708223842824309167, 4378483388165924164886086401654804541)
(5781908735448751194342417, 2282948607239853703861815471121478041)
(5805311356758763706695917, 1415021355465625977424510976276679541)
(671077204855498227518997, 32059330489755416601952661436907512061)
(11989358974451671095290917, 7676251532810749960821688176141854541)
(1562293871267413317586917, 28624141413758987900969794922124054541)
(1576322062238846728594867, 28567614529331320357493343879981604341)
(2645057615249834709978417, 24010530256918791161702085809168042041)
(3720489485435437102441417, 18807298122579603431575117122003441541)
(3816536637918617509199667, 18301982965535924241809674496531960791)
(4512681870129300042673917, 14342071221839135668437887977558521541)
(4578423758711455665926277, 13933193381599661569174839553353688141)
(-4756917288381900348953883, 47019904101052608363476683373565059341)
(-7423289545094572164623783, 50434409310156378282912115980862469541)
(13282176913295608720023909, 19267088965938182571829569054014801293)
(15542115915720838350209667, 35282145912077658427145516554287410791)
(-17329093872184303283521083, 10239155062825108632969670098428466541)

Commentary

specialization at T = -12396/65165 of the rank-9 fibration of X720 given by: y^2 + (504T^2 - 8334T - 1499)xy = x(x + (445T - 87)(127T - 4)(25T + 3)(89T + 168))(x + (583T - 196)(T - 13)(5T + 22)(253T + 29)).

last edited by wgxli at 2026-10-09 05:31:20 UTC · history