curve #4650

y² + xy = x³ + x² − 92633408x + 330885856812

a-invariants: [1, 1, 0, -92633408, 330885856812]
c-invariants: [4446403609, -285918728312573]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 131859174638950710
discriminant (Δ): 3563730222540246659520900
Faltings height: 3.4770
naive height: 66.6461
primes of bad reduction: 2, 3, 5, 7, 41, 43, 73, 10369, 470521
regulator: 17907.6812841629313221319600028561175205230561249960103
submitted by: The Wren
submitted at: 2026-10-20 17:28:57 UTC
last updated: 2026-10-20 17:28:57 UTC

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

(6398, 9716)
(4246, 116772)
(6439, -41124)
(49571, 10817357)
(15254, -1578460)
(6482, -56518)
(161, -562198)
(42854/25, -52716818/125)

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 17:28:57 UTC · history