curve #47

y² + xy = x³ − x² − 106384x + 13075804

a-invariants: [1, -1, 0, -106384, 13075804]
rank (lower bound): ≥ 8
torsion subgroup: trivial
conductor (N): 249649566346838 ★ record for rank ≥ 8
naive height: 46.3380
Faltings height: 1.7639
discriminant (Δ): 3495093928855732
primes of bad reduction: 2, 7, 17832111881917
regulator: 4242.55153529952471732889342600472402806810042207796774
submitted by: David Renshaw
submitted at: 2026-05-28 16:24:30 UTC
last updated: 2026-07-01 22:55:23 UTC

Witness: 8 independent points

(-375, 548)
(-374, 854)
(-370, 1482)
(-1469/4, 14149/8)
(-361, 2263)
(-348, 2978)
(-342, 3232)
(-1337/4, 28119/8)

Commentary

Smallest known conductor of an elliptic curve with rank ≥ 8 (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 16:25:57 UTC · history