curve #4652

y² + xy = x³ + x² − 117613367x + 128515058625

a-invariants: [1, 1, 0, -117613367, 128515058625]
c-invariants: [5645441641, -111079351464245]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 490807272278185854
discriminant (Δ): 96983398226809633429423332
Faltings height: 3.6775
naive height: 67.3623
primes of bad reduction: 2, 3, 37, 73, 89, 137, 937, 36313
regulator: 46770.6230387883523222073384849405572321083161330168197
submitted by: The Wren
submitted at: 2026-10-20 17:34:31 UTC
last updated: 2026-10-20 17:34:31 UTC

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

(-2744, 657569)
(-1943, 592332)
(112, 339567)
(249, 314907)
(-65167/64, -254311283/512)
(906, 150219)
(371, 291245)
(461391, 313086729)

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 17:34:31 UTC · history