curve #1958

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y² = x³ + x² − 6975831472259069205275460x + 8982736568693575960378824754803868608

a-invariants

[0, 1, 0, -6975831472259069205275460, 8982736568693575960378824754803868608]

rank (lower bound)

≥ 19

torsion subgroup

trivial

conductor (N)

83381005700353763816706043731041708749035035649707435573880

discriminant (Δ)

-13132481663695722220075359768828762423367212917596484383354832432708508000000

Faltings height

13.3169

naive height

183.6999

primes of bad reduction

2, 3, 5, 7, 23, 431, 415553, 61915409327, 389186320132092517280729790246779969

regulator

87135257878425915.47915851885076405277272554564022128456655773571533425716571

submitted by

Ritik Jain

submitted at

2026-09-25 20:06:42 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 20:07:57 UTC

Witness: 19 independent points log in to add more points →

(-47281787933694189711634292290998/22785799631862706609, -414501300043231244756596543883196483918257091834/108766814938703411322146655977)
(-1670645055834, -3996750286745013210)
(-6939451822590096/4489, -1205766185919934850038800/300763)
(5299897842678, -10994541089396082858)
(80107645620459/49, -478705431495892748580/343)
(548646589646742/529, -20588618401002392001894/12167)
(-85733719901646/289, -16313864026793309194050/4913)
(-72384570103506/25, -276917901222134598678/125)
(-4900371233848292796/2640625, 16914032492027320024800466608/4291015625)
(-176598592041746586/237169, 428510923042367900666164170/115501303)
(111673011618429/100, 1607878770943430069133/1000)
(424682134599024/169, 5944677242224617914640/2197)
(-181665463841155851/62500, -33881110912039644271762857/15625000)
(5104385233448799/4225, -418148759544204635012682/274625)
(24623983458246/25, -218924198299057100094/125)
(173286347297427842958/161823841, -3407954880156107346816217478718/2058561081361)
(470350503308316704109/71605444, 9514734944746543725761195662875/605925267128)
(96347465955577039847366274834/146070727157152009, 120623698864008574447075382806187016283202790/55827063791858422095184027)
(9142854392256664789386336/39979178222281, 687620457260182755097366764057991869960/252784705794264535429)

Commentary

Specialization at T = 5 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 = 132987T^2 + 93270T + 585575, p = -48077T^2 + 65376T + 319973, q = 13974T^2 - 848745T + 719796, D = 3762979682T^4 + 7417401493T^3 - 91707153902T^2 - 10715099623T + 249952800570, E = 2431655842T^4 - 17839413059T^3 + 73246802384T^2 + 439113416444T + 183933292704.

last edited by wgxli at 2026-09-26 02:37:23 UTC · history