curve #4658

y² = x³ − x² − 134175x + 4217444631

a-invariants: [0, -1, 0, -134175, 4217444631]
c-invariants: [6440416, -3643833518720]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 321457482575232
discriminant (Δ): -7683596973611678400768
Faltings height: 2.8783
naive height: 57.8481
primes of bad reduction: 2, 3, 17, 47, 89, 263, 2633
regulator: 9408.338452254765990094749639113867694621533701360600
submitted by: The Wren
submitted at: 2026-10-20 17:59:33 UTC
last updated: 2026-10-20 17:59:33 UTC

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

(1383, 81702)
(-130, 65059)
(-931, 59452)
(2807, 161090)
(-1554, 25899)
(10461, 1071204)
(1205, 76184)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Seven 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 17:59:33 UTC · history