curve #4251

y² = x³ + x² − 375446152299243359272119409x + 8004421081464486742085823923258618005535

a-invariants: [0, 1, 0, -375446152299243359272119409, 8004421081464486742085823923258618005535]
c-invariants: [18021415310363681245061731648, -6915819814385424673654014051782916327172096]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 81227180467392566378365208705376
discriminant (Δ): -24291506509440501984862372559700708847292508840321386491148400337200279984231518208
Faltings height: 14.4935
naive height: 197.2848
primes of bad reduction: 2, 3, 11, 13, 17, 31, 43, 89, 97, 167, 683, 733, 1031, 12611, 27823
regulator: 822485258499.8708949900536403050358743340435729145543396349
submitted by: Michael Rubinstein
submitted at: 2026-10-15 18:12:40 UTC
last updated: 2026-10-15 18:12:40 UTC

Witness: 10 independent points log in to add more points →

(97052251757015, 941124450631724927700)
(-355587895361966128897/15062161, 3560696613863736725643512663988/58456246841)
(-17138841305405377/841, 2068333791764758434470268/24389)
(846987112788604583/69169, 1317230633071004500854241524/18191447)
(9131265231190897217/896809, 61461168540317266393280603556/849278123)
(88002217373574349151/3367225, 782625868612195497723443201724/6178857875)
(274706952621478664621/4674244, 4392551294133509262344301776349/10105715528)
(540579137861245339607/22934521, 12156357425911616747238983791932/109833421069)
(1258662524772650177981/7856809, 44370304039407808421039458522584/22022635627)
(-9790612520652939115/418609, 17111418320999582771533376400/270840023)

Commentary

Specialization at t = -167/1118 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from the four generic sections of the fibration and a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-10-15 18:12:40 UTC · history