curve #48

y² + xy = x³ − x² − 135004x + 97151644

a-invariants: [1, -1, 0, -135004, 97151644]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 32107342006814614 ★ record for rank ≥ 9
naive height: 50.3060
Faltings height: 2.2491
discriminant (Δ): -3917095724831382908
primes of bad reduction: 2, 61, 263174934482087
regulator: 33288.740573083599055593661213206843477412693238540465426
submitted by: David Renshaw
submitted at: 2026-05-28 18:27:55 UTC
last updated: 2026-07-01 22:51:30 UTC

Witness: 9 independent points

(-556, -42)
(-452, -7878)
(-233, -10650)
(-25, -10013)
(197, 8743)
(446, 10980)
(801, 22015)
(1219, 41138)
(1876, 79334)

Commentary

Smallest known conductor of an elliptic curve with rank ≥ 9 (Elkies–Watkins, 2004). From their paper "Elliptic Curves of Large Rank and Small Conductor" (arXiv:math/0403374).

last edited by David Renshaw at 2026-05-28 18:41:34 UTC · history