curve #57

y² + y = x³ − 1387x + 68046

a-invariants: [0, 0, 1, -1387, 68046]
c-invariants: [66576, -58791960]
rank (lower bound): ≥ 7
torsion subgroup: trivial
conductor (N): 1829517077483
discriminant (Δ): -1829517077483 ★ record for rank ≥ 7
Faltings height: 1.0365 ★ record for rank ≥ 7
naive height: 35.7790 ★ record for rank ≥ 7
primes of bad reduction: 3527, 518717629
regulator: 1306.442211994994493978784223406033003476005702575198
submitted by: David Renshaw
submitted at: 2026-05-29 00:04:00 UTC
last updated: 2026-07-01 22:56:22 UTC

Witness: 7 independent points log in to add more points →

(-49, 135)
(-47, 171)
(-44, 209)
(-43, 219)
(-42, 228)
(-34, 275)
(-30, 287)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=7: [0,0,1,-1387,68046], |Delta|=1829517077483; their I-column records 71 integral x-coordinates. The submitted points certify rank >= 7 here.

last edited by David Renshaw at 2026-05-29 00:13:05 UTC · history