curve #4664

y² + xy + y = x³ − 164323x + 16354602

a-invariants: [1, 0, 1, -164323, 16354602]
c-invariants: [7887481, -14142207565]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 1294832923826
discriminant (Δ): 168227283129321572
Faltings height: 2.0073
naive height: 47.6424
primes of bad reduction: 2, 13, 17, 19, 263, 313, 1873
regulator: 730.87447064113706851126953635030064445476313602489
submitted by: The Wren
submitted at: 2026-10-20 18:13:53 UTC
last updated: 2026-10-20 18:13:53 UTC

Witness: 6 independent points log in to add more points →

(-419, 3628)
(-156, 6258)
(633, 12570)
(6945, 574338)
(3847/4, 214965/8)
(700/9, 53246/27)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Six 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 18:13:53 UTC · history