curve #4689

y² + xy = x³ + x² − 46913381x + 123653940825

a-invariants: [1, 1, 0, -46913381, 123653940825]
c-invariants: [2251842313, -106853893690085]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 1065080179392582
discriminant (Δ): 505467881696490350724
Faltings height: 3.0502
naive height: 64.6050
primes of bad reduction: 2, 3, 19, 23, 73, 181, 577, 53281
regulator: 6680.486654247187310127681572040305427208665843252137
submitted by: The Wren
submitted at: 2026-10-20 20:42:44 UTC
last updated: 2026-10-20 20:42:44 UTC

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

(4848, 98523)
(3793, 15389)
(4031, 6107)
(3100, 88035)
(7413, 424449)
(3407/4, 2319547/8)
(-5800, -445075)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. I checked the seven supplied rational points exactly on this minimal model. The source records rank 7 with independent points and a matching 2-descent upper bound. Rational torsion Z/2Z. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 20:42:44 UTC · history