curve #3712

y² + xy + y = x³ − 19307510245957239x + 1040321745339454020879686

a-invariants: [1, 0, 1, -19307510245957239, 1040321745339454020879686]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 79182482567274710
discriminant (Δ): -6903378250880291137646188042617871992304351407100
Faltings height: 8.1960
naive height: 124.1263
primes of bad reduction: 2, 5, 13, 17, 23, 31, 37, 103, 113, 116689
regulator: 260463.85344491051502691726790239293322824620586005
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:48:01 UTC
last updated: 2026-09-30 01:48:01 UTC

Witness: 6 independent points log in to add more points →

(289094507/4, 1197511386379/8)
(80408068, 87806756173)
(2398296433, 117256288642278)
(56837482/49, 346060724320744/343)
(145782519467/1849, 7180884275205466/79507)
(93500791458568/1256641, 176227970218951431293/1408694561)

Commentary

Specialization at t = -27223/14911 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 01:48:02 UTC · history