curve #3637

y² + xy + y = x³ + x² − 78799314921090126336635x + 8509215482293751203148041509575561

a-invariants: [1, 1, 1, -78799314921090126336635, 8509215482293751203148041509575561]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 455514655953569565656462935595492873046
discriminant (Δ): 34915865618761763073549337830972982408707179426408720321728921731072
Faltings height: 11.9006
naive height: 169.7772
primes of bad reduction: 2, 3, 7, 11, 23, 37, 43, 71, 157, 631, 12721, 1299673, 11326561, 20456041
regulator: 520589953403783.4570444427791043737527034655893209553375956790392
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:24:17 UTC
last updated: 2026-09-29 19:24:17 UTC

Witness: 13 independent points log in to add more points →

(136975485403, 16900049119712534)
(-21342102184229/121, 173135843458133284306/1331)
(-15783081350461/49, 7453908939379158698/343)
(12809124036063/169, 119774482860609802438/2197)
(160221606163411/225, 1889657427228542709946/3375)
(5860864089980091/37249, 17620391238430943288302/7189057)
(5875730590422265/136161, 3619324592691474753612928/50243409)
(10835622926510413/69169, 55622798740791208173538/18191447)
(11776358776075347/52441, 555310088664689933554822/12008989)
(108022578708368691/410881, 20339919858737708931107942/263374721)
(304675932725825699/5329, 168171334476281740427089654/389017)
(2908388942949335179/19193161, 582928367452960545245988734/84085238341)
(4015906108044053039/23863225, 440791490872502887069780642/116571854125)

Commentary

Specialization at T = -265/63 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:24:17 UTC · history