curve #101

y² + xy = x³ − x² − 4513546x + 3716615296

a-invariants: [1, -1, 0, -4513546, 3716615296]
rank (lower bound): ≥ 10
conductor (N): 39432942782223365758
naive height: 57.5947
Faltings height: 2.6471
discriminant (Δ): -78865885564446731516
primes of bad reduction: 2, 73523, 268167395115973
regulator: 626053.0026059040614370288739650539139511584443021140154468
submitted by: David Renshaw
last updated: 2026-07-01 22:50:31

Witness: 10 independent points

(1216, 4400)
(1250, 4536)
(1254, 4606)
(1174, 5310)
(1286, 5500)
(1154, 6040)
(1312, 6552)
(1320, 6916)
(1434, 13120)
(942, 16852)

Commentary

EW 2004 (Elkies-Watkins, ANTS-VI, arXiv:math/0403374) Table 2 low-conductor rank-10 list; this curve was absent from the leaderboard. Rank-10 witness via 10 independent rational points (Neron-Tate pairing).

last edited by David Renshaw at 2026-06-23 23:50:32 · history