curve #3719

y² + xy + y = x³ − x² − 189344178381656298886178x + 31712176383204412199788762803759137

a-invariants: [1, -1, 1, -189344178381656298886178, 31712176383204412199788762803759137]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 1693477477526556829770
discriminant (Δ): 694031916904332139240570697877480993938759864943604300161945600
Faltings height: 11.7874
naive height: 172.4072
primes of bad reduction: 2, 3, 5, 13, 19, 37, 43, 61, 67, 97, 103, 271, 4327
regulator: 1989258.346585550007553568637719820727099851379880463
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:49:26 UTC
last updated: 2026-09-30 01:49:26 UTC

Witness: 7 independent points log in to add more points →

(-62762872737, 208203601991477023)
(12309612294897/49, 2334218148590089/343)
(23052923744937/289, 642761138977130558519/4913)
(8094611808468487/33124, 35704465284930450150689/6028568)
(3697564874076033/14641, 2035069667637663602503/1771561)
(70823250272805297/271441, 1197207014817343925044295/141420761)
(3894516757287338977/15499969, 1724423215946297923198239/61023377953)

Commentary

Specialization at t = -1737309/17308 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), 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 2026-09-30 01:49:26 UTC · history