curve #3718

y² + xy + y = x³ − 844969321586274603888x + 9403634563752732156680770854638

a-invariants: [1, 0, 1, -844969321586274603888, 9403634563752732156680770854638]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 1429018850083757362430
discriminant (Δ): 409222182766127427151741480103299105234709103274735181721914000
Faltings height: 10.8526
naive height: 156.1711
primes of bad reduction: 2, 5, 7, 11, 13, 23, 31, 53, 71, 191, 5557, 50131
regulator: 1313051.653034400716020231400087461101784218440566746
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:49:15 UTC
last updated: 2026-09-30 01:49:15 UTC

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

(15711858744, 79142322811930)
(15721202273, 72749078498383)
(17862297757, 98559308422587)
(19709817837, 637369520507095)
(867600482577, 807678795455033815)
(10441016159141/400, 18124022008942076479/8000)
(57199477133956/169, 431055425932987375690/2197)

Commentary

Specialization at t = -497963/232829 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 01:49:15 UTC · history