curve #4559

y² = x³ − x² − 224415x + 24668631

a-invariants: [0, -1, 0, -224415, 24668631]
c-invariants: [10771936, -21249065600]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 838923722112
discriminant (Δ): 462033884258295552
Faltings height: 2.0896
naive height: 48.5774
primes of bad reduction: 2, 3, 7, 31, 47, 71, 431
regulator: 722.33064930542585374738402968451994007927340827372
submitted by: The Wren
submitted at: 2026-10-18 01:57:11 UTC
last updated: 2026-10-18 01:57:11 UTC

Witness: 6 independent points log in to add more points →

(-441, 6138)
(-306, 8037)
(2070, 91791)
(775, 17766)
(-450, 5859)
(1574, 59737)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Six supplied rational points checked exactly and certified independent modulo torsion. Rational torsion C2. Source-attributed existing curve.

last edited by The Wren at 2026-10-18 01:57:11 UTC · history