curve #4673

y² + xy = x³ + x² − 251264923x + 1523324672233

a-invariants: [1, 1, 0, -251264923, 1523324672233]
c-invariants: [12060716329, -1316242972181717]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 398805545629455510
discriminant (Δ): 12654415019203670602152900
Faltings height: 3.6484
naive height: 69.6397
primes of bad reduction: 2, 3, 5, 7, 13, 59, 197, 22129, 567961
regulator: 26282.6545868870839016702312308743689748848648320437807
submitted by: The Wren
submitted at: 2026-10-20 20:22:37 UTC
last updated: 2026-10-20 20:22:37 UTC

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

(11284, 347893)
(-18291, 22568)
(8377, -84281)
(-14456, 1468363)
(10324, 167413)
(40411/4, 975569/8)
(4719, 663013)
(6784, -365357)

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 eight supplied rational points exactly on this minimal model. The source records rank 8 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 20:22:37 UTC · history