curve #4654

y² + xy = x³ + x² − 159971236x + 719912052700

a-invariants: [1, 1, 0, -159971236, 719912052700]
c-invariants: [7678619353, -622061603177885]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 495488867460190302
discriminant (Δ): 38067092677556711716837284
Faltings height: 3.6502
naive height: 68.2851
primes of bad reduction: 2, 3, 73, 79, 97, 109, 457, 1487, 1993
regulator: 97498.5942909570826620368261395945116613513442408033274
submitted by: The Wren
submitted at: 2026-10-20 17:37:30 UTC
last updated: 2026-10-20 17:37:30 UTC

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

(-38, 852070)
(4423, 312289)
(11858, -706306)
(4618, 279958)
(126357, -44761655)
(10371, -425263)
(-37517/9, 31010411/27)
(-2729, 1067272)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Eight supplied rational points checked exactly on the displayed model. Rational torsion Z/2Z. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 17:37:31 UTC · history