curve #72

y² + y = x³ − 7x + 36

a-invariants: [0, 0, 1, -7, 36]
rank (lower bound): ≥ 4
conductor (N): 545723
naive height: 20.7040 ★ record for rank ≥ 4
Faltings height: -0.2177 ★ record for rank ≥ 4
discriminant (Δ): -545723
primes of bad reduction: 545723
regulator: 3.306378474591305516092927336098302768950667315
submitted by: David Renshaw
last updated: 2026-07-01 23:03:09

Witness: 4 independent points

(-4, 0)
(-3, 5)
(-2, 6)
(-1, 6)

Commentary

Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=336, c6=-31320, Delta=-545723 with 545723 prime. The submitted points (-4,0), (-3,5), (-2,6), and (-1,6) 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 2026-06-01 13:37:21 · history