curve #4692

y² + xy + y = x³ − 264147693x + 1652389636952

a-invariants: [1, 0, 1, -264147693, 1652389636952]
c-invariants: [12679089241, -1427683664960605]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 1294885748254202
discriminant (Δ): 529111034908639004432
Faltings height: 3.3497
naive height: 69.7897
primes of bad reduction: 2, 13, 17, 53, 61, 113, 157, 3929
regulator: 22388.22306212235468213469021951528243802948797696061
submitted by: The Wren
submitted at: 2026-10-20 20:45:07 UTC
last updated: 2026-10-20 20:45:07 UTC

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

(9335, 3418)
(9270, 14338)
(9391, -3570)
(9371, -2702)
(9374, -6137)
(9400, -1977)
(9378, -4025)

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 20:45:08 UTC · history