curve #4686

y² = x³ + x² − 376855x + 88455789

a-invariants: [0, 1, 0, -376855, 88455789]
c-invariants: [18089056, -76534336000]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 817578526976
discriminant (Δ): 35582652651049472
Faltings height: 2.0152
naive height: 50.1325
primes of bad reduction: 2, 17, 47, 89, 97, 463
regulator: 1358.8077018147067950661820049355933863927392926033
submitted by: The Wren
submitted at: 2026-10-20 20:38:55 UTC
last updated: 2026-10-20 20:38:55 UTC

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

(-593, 10186)
(239, 3478)
(383, 670)
(709, 13348)
(-513, 12126)
(55061/100, 6941759/1000)

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 six supplied rational points exactly on this minimal model. The source records rank 6 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 20:38:56 UTC · history