curve #4663

y² + xy = x³ + x² − 6414202x + 4618561600

a-invariants: [1, 1, 0, -6414202, 4618561600]
c-invariants: [307881721, -3992746335245]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 568648314593502
discriminant (Δ): 7663449288340676614212
Faltings height: 2.9080
naive height: 58.6357
primes of bad reduction: 2, 3, 11, 13, 113, 139, 313, 1193
regulator: 3350.593623037901708739610180389158399173869955266249
submitted by: The Wren
submitted at: 2026-10-20 18:11:41 UTC
last updated: 2026-10-20 18:11:41 UTC

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

(-2258, 88282)
(-1686, 104012)
(-2138, 93616)
(2043, 5815)
(574, 33274)
(3286, 136330)
(2051, 8774)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Seven 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 18:11:41 UTC · history