curve #71

y² = x³ − x² − 75x − 134

a-invariants: [0, -1, 0, -75, -134]
rank (lower bound): ≥ 4
torsion subgroup: trivial
conductor (N): 16430032
naive height: 24.5794
Faltings height: 0.0861
discriminant (Δ): 16430032
primes of bad reduction: 2, 353, 2909
regulator: 29.93011794279284801770139339930493926616246710
submitted by: David Renshaw
submitted at: 2026-05-29 13:46:05 UTC
last updated: 2026-07-01 23:03:09 UTC

Witness: 4 independent points

(-2, 2)
(-6, 8)
(10, 4)
(30, 154)

Commentary

Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=3616, c6=137440, Delta=16430032 = 2^4*353*2909. The submitted points (-2,2), (-6,8), (10,4), (30,154) 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 13:46:08 UTC · history