curve #1184

y² + xy = x³ − 267568337358391229485x + 1684853939243502235524622982225

a-invariants: [1, 0, 0, -267568337358391229485, 1684853939243502235524622982225]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/10ℤ
conductor (N): 2914314327210 ☆ record for ℤ/10ℤ torsion, rank ≥ 3
discriminant (Δ): -350454986086279087597351434974996124682603065833747251200000
Faltings height: 10.4337
naive height: 152.7216
primes of bad reduction: 2, 3, 5, 11, 13, 19, 29, 109, 11311
regulator: 14948.569527002120086478243892711430957094223
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:40 UTC
last updated: 2026-09-23 21:38:40 UTC

Witness: 3 independent points log in to add more points →

(1721568508570417730/177235969, 113558364381360549042949855/2359542455297)
(3310921544308211650/194351481, 3898844358963966781829609635/2709453996621)
(496049677212875691106/53301418641, 342285594444255790379139807577/12305751823066311)

Commentary

Obtained by computing a rational isogeny from curve#1091 and mapping all 3 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:38:41 UTC · history