curve #157

y² + xy = x³ − x² − 227292004x + 882331831684

a-invariants: [1, -1, 0, -227292004, 882331831684]
rank (lower bound): ≥ 12
conductor (N): 12212741842431104280769438 ★ record for rank ≥ 12
naive height: 69.3388 ★ record for rank ≥ 12
Faltings height: 3.8113 ★ record for rank ≥ 12
discriminant (Δ): 415233222642657545546160892 ★ record for rank ≥ 12
primes of bad reduction: 2, 17, 8046817087, 44638555294561
regulator: 42564662.256038750630862983698271383495644865238969730938118940
submitted by: Andrew Sutherland
last updated: 2026-07-01 22:46:37

Witness: 12 independent points

(117790/9, -10607936/27)
(1309577033/73441, 31285012576920/19902511)
(1881346627/18769, 80581143046275/2571353)
(2591120/961, 15977490022/29791)
(-91616894/5625, -213077007104/421875)
(37120, 6583838)
(15910, -1145142)
(13305206/841, 27031784336/24389)
(-57830/49, 367859024/343)
(34256, -5787210)
(17920627/1369, 19285307426/50653)
(305109061/24336, 192990890929/3796416)

Commentary

Found by Noam D. Elkies in June 2026, using an improved version of the methods described in his JMM 2023 talk https://abel.math.harvard.edu/~elkies/Elkies_JMM23.pdf

last edited by Andrew Sutherland at 2026-06-24 23:40:21 · history