curve #61

y² + y = x³ + 752558524241981430

a-invariants: [0, 0, 1, 0, 752558524241981430]
rank (lower bound): ≥ 13
conductor (N): 81553583866934729091924419700360328569
naive height: 95.8477
Faltings height: 6.0450
discriminant (Δ): -244660751600804187275773259101080985707
primes of bad reduction: 3, 190669, 15787747861309
regulator: 671098787666.6903251127775215461325395089393390179049329444143585
submitted by: David Renshaw
last updated: 2026-07-01 22:45:37

Witness: 13 independent points

(-3635871/4, -314862269/8)
(-3627871/4, -644755819/8)
(-3436375/4, -2754036385/8)
(-856990, -350938406)
(-845232, -385629747)
(-3332823/4, -3338220641/8)
(-792924, -504008563)
(-783172, -521721587)
(-2967655/4, -4693365385/8)
(-2538855/4, -5639045385/8)
(-521974, -781244603)
(-120210, -866499531)
(894450, -1211674470)

Commentary

Provenance: Noam D. Elkies, 2009 rank-13 Mordell curve. Elkies originally gave the model y^2 = x^3 + 48163745551486811536 with 13 independent integral points; this database stores the global minimal model y^2 + y = x^3 + 752558524241981430, with the witness points transformed accordingly. Andrew Sutherland's MIT rank-record page lists the original curve and points.

last edited by David Renshaw at 2026-06-25 20:11:18 · history