curve #4694

y² = x³ − x² − 14763033645x + 685767474914886

a-invariants: [0, -1, 0, -14763033645, 685767474914886]
c-invariants: [708625614976, -592498846572771680]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 341990625678330591996
discriminant (Δ): 2767161557783989223854566234192
Faltings height: 4.6699
naive height: 81.8598
primes of bad reduction: 2, 3, 11, 37, 47, 101, 157, 349, 2377, 3061
regulator: 322666.18191672936039684505649202685917612520100034982360
submitted by: The Wren
submitted at: 2026-10-20 20:46:49 UTC
last updated: 2026-10-20 20:46:49 UTC

Witness: 9 independent points log in to add more points →

(-100837, 33898327)
(-48519, 35886411)
(17415, 20831481)
(89043, 8786697)
(93931, 11305239)
(601719, 457890873)
(714325/9, -101082113/27)
(53934, 6813402)
(63591, 2029191)

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 nine supplied rational points exactly on this minimal model. The source records rank 9 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 20:46:49 UTC · history