curve #59

y² + y = x³ − 167419x + 30261330

a-invariants: [0, 0, 1, -167419, 30261330]
c-invariants: [8036112, -26145789336]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 95276302704064331
discriminant (Δ): -95276302704064331 ★ record for rank ≥ 9
Faltings height: 1.9827 ★ record for rank ≥ 9
naive height: 47.9739
primes of bad reduction: 95276302704064331
regulator: 294208.68480812036499615812853149590022566024193525846092
submitted by: David Renshaw
submitted at: 2026-05-29 00:04:01 UTC
last updated: 2026-07-01 22:51:30 UTC

Witness: 9 independent points log in to add more points →

(-480, 174)
(-479, 743)
(-478, 1035)
(-477, 1260)
(-463, 2919)
(-439, 4376)
(-422, 5075)
(-384, 6158)
(-347, 6824)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-167419,30261330], |Delta|=95276302704064331; their I-column records 135 integral x-coordinates. The submitted points certify rank >= 9 here.

last edited by David Renshaw at 2026-05-29 00:13:07 UTC · history