curve #3494

y² + xy + y = x³ − 2612439389653453x + 103699340708735815532756

a-invariants: [1, 0, 1, -2612439389653453, 103699340708735815532756]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 23340040158996330
discriminant (Δ): -3504448328079385480608964470344818871253999937500
Faltings height: 8.0051
naive height: 119.5147 ☆ record for ℤ/6ℤ torsion, rank ≥ 6
primes of bad reduction: 2, 3, 5, 7, 11, 17, 37, 227, 1201, 58921
regulator: 5497.1822864696564248373510912814768720271278889714
submitted by: jesper-petersen
submitted at: 2026-09-27 10:57:18 UTC
last updated: 2026-09-27 10:57:18 UTC

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

(212804215/4, 2716167473161/8)
(202535671, 2807520819644)
(359311405, 6749176323872)
(851679685/16, 21745371395333/64)
(5921980345/256, 966168902510597/4096)
(11103921025/49, 1145226165790286/343)

Commentary

Torsion Z/6 specialization of the universal Tate family y^2 + t*x*y + (t+2)*y = x^3 at t = -8431/8415. Found in a conductor-oriented search of reduced rational parameters, using bad-reduction support filtering followed by Mestre-Nagao screening and exact PARI/GP descent. PARI/GP 2.17.4 gives rank bounds [6,6] and torsion Z/6Z.

last edited by jesper-petersen at 2026-09-27 10:57:18 UTC · history