curve #3491

y² = x³ + x² − 948152134593796x + 11237381939628169443140

a-invariants: [0, 1, 0, -948152134593796, 11237381939628169443140]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/6ℤ
conductor (N): 8562359590620
discriminant (Δ): 17278187955776727061097576436480 ☆ record for ℤ/6ℤ torsion, rank ≥ 5
Faltings height: 6.7880
naive height: 115.0702
primes of bad reduction: 2, 3, 5, 11, 23, 53, 727, 14639
regulator: 13789.4368318126082621190765393275902576800533130
submitted by: jesper-petersen
submitted at: 2026-09-27 10:33:31 UTC
last updated: 2026-09-27 10:33:31 UTC

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

(71186885/4, 1104781425/8)
(436885724/25, 275255902482/125)
(5137773896/289, 1810873578/4913)
(1049840135971/59049, 138331479685058/14348907)
(63975424543844/3598609, 278891480602482/6826561273)

Commentary

Rank-5 Z/6Z curve obtained via the 2-isogeny involution w_2(t) = -(2t+12)/(t+2) from the specialization t = -7336/3657, giving the isogenous parameter t = 14606/11. Found during a conductor/discriminant-first search of the universal X_1(6) family. PARI/GP 2.17.4 gives rank bounds [5,5] and torsion Z/6Z.

last edited by jesper-petersen at 2026-09-27 10:33:31 UTC · history