curve #69

y² + xy = x³ + 1

a-invariants: [1, 0, 0, 0, 1]
rank (lower bound): ≥ 2
conductor (N): 433
naive height: 13.5255 ★ record for rank ≥ 2
Faltings height: -0.8152 ★ record for rank ≥ 2
discriminant (Δ): -433
primes of bad reduction: 433
regulator: 0.224694163418166741612071133995500441366829
submitted by: David Renshaw
last updated: 2026-07-01 23:04:07

Witness: 2 independent points

(0, 1)
(-1, 1)

Commentary

Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=1, c6=-865, Delta=-433. The submitted points (0,1), (-1,1) were checked exactly on the curve, with an additional finite-field independence sanity check; the site Neron-Tate verifier gives the definitive certificate.

last edited by David Renshaw at 2026-05-29 13:46:07 · history