curve #89

y² = x³ − 6529826212x + 195995662948900

a-invariants: [0, 0, 0, -6529826212, 195995662948900]
rank (lower bound): ≥ 13
conductor (N): 38252643743425234347938185256
naive height: 79.4125
Faltings height: 4.5399
discriminant (Δ): 1224084599789607499134021928192
primes of bad reduction: 2, 2741, 300427, 5806620890419143251
regulator: 3016078366.978183471091966719699927484854209920272441966848140771
submitted by: David Renshaw
last updated: 2026-07-01 22:44:39

Witness: 13 independent points

(-104, 14024078)
(-1032, 14238446)
(2022, 13520378)
(-2566, 14585414)
(3616, 13131302)
(-4471, 15003371)
(-4574, 15025558)
(5193/4, 109550501/8)
(5258, 12720346)
(-6047/4, 114783601/8)
(6337, 12444703)
(-6694, 15472762)
(-11031, 16330459)

Commentary

Smallest known conductor (N = 38252643743425234347938185256) elliptic curve of rank >= 13, announced by Noam D. Elkies in 'New elliptic curves of large rank and low conductor' (preliminary report, JMM Special Session on arithmetic geometry informed by computation, Boston, January 4, 2023). Witness points found by ellratpoints and certified independent by Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 16:55:32 · history