curve #3618

y² + xy + y = x³ − 29499173172906148x + 8316765760147324691627078

a-invariants: [1, 0, 1, -29499173172906148, 8316765760147324691627078]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 10817942205250900617851910
discriminant (Δ): -28237938217034303011063296998709996320882204124592500
Faltings height: 8.7466
naive height: 128.2838
primes of bad reduction: 2, 3, 5, 11, 19, 61, 83, 131, 191, 229, 349, 359, 479, 991
regulator: 27523467486.63984306211591136005078339963832103979406588762
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:20:36 UTC
last updated: 2026-09-29 19:20:36 UTC

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

(29130763, 2735337832913)
(112851499, 2534693206115)
(172231504, 2888702716130)
(459906724, 9592824727985)
(25147710946/25, 3945808286721691/125)
(26821198513/49, 4286317835939411/343)
(30421158913/529, 31751877436170379/12167)
(103423438693951/124609, 1036793371170365156275/43986977)
(2788252446514099/1371241, 146775891974810817532385/1605723211)
(311790935011846561/49729, 174098544242564675071780715/11089567)

Commentary

Specialization at T = 26 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 19:20:36 UTC · history