curve #58

y² + y = x³ + x² − 16440x + 1394010

a-invariants: [0, 1, 1, -16440, 1394010]
c-invariants: [789136, -1209159640]
rank (lower bound): ≥ 8
torsion subgroup: trivial
conductor (N): 561715239383323
discriminant (Δ): -561715239383323
Faltings height: 1.5245
naive height: 41.8264 ★ record for rank ≥ 8
primes of bad reduction: 23382011, 24023393
regulator: 49900.6667347885574015353067843339561342699573634387980
submitted by: David Renshaw
submitted at: 2026-05-29 00:04:00 UTC
last updated: 2026-07-01 22:54:25 UTC

Witness: 8 independent points log in to add more points →

(-159, 116)
(-151, 675)
(-140, 985)
(-138, 1026)
(-124, 1241)
(-114, 1341)
(-93, 1458)
(-91, 1464)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [0,1,1,-16440,1394010], |Delta|=561715239383323; their I-column records 84 integral x-coordinates. The submitted points certify rank >= 8 here.

last edited by David Renshaw at 2026-05-29 00:13:06 UTC · history