curve #3490

y² + xy + y = x³ − 2376745793554x + 516874740970797596

a-invariants: [1, 0, 1, -2376745793554, 516874740970797596]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/6ℤ
conductor (N): 591998388510 ☆ record for ℤ/6ℤ torsion, rank ≥ 5
discriminant (Δ): 743854093030240479885720191052127274280
Faltings height: 6.1515
naive height: 97.1039
primes of bad reduction: 2, 3, 5, 23, 31, 103, 167, 1609
regulator: 48261.1883179379852781996820613634028508079492542
submitted by: jesper-petersen
submitted at: 2026-09-27 10:31:45 UTC
last updated: 2026-09-27 10:31:45 UTC

Witness: 5 independent points log in to add more points →

(5894749735/4, 452576670634201/8)
(-710290010/841, 33820301854453/24389)
(206837986/1849, 39939448348439/79507)
(5107229935/484, 361096172573651/10648)
(81402874726/50625, 10508466945418424/11390625)

Commentary

Torsion Z/6 specialization of the universal Tate family y^2 + t*x*y + (t+2)*y = x^3 at t = -9914/3841. Found in a conductor-first search over reduced rational parameters using bad-reduction support filtering, Mestre-Nagao scoring, and exact PARI/GP 2-descent. PARI/GP 2.17.4 gives rank bounds [5,5] and torsion Z/6Z.

last edited by jesper-petersen at 2026-09-27 10:31:45 UTC · history