curve #4674

y² + xy = x³ + x² − 254430033x + 1550489097537

a-invariants: [1, 1, 0, -254430033, 1550489097537]
c-invariants: [12212641609, -1339714175083973]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 423625044939673170
discriminant (Δ): 15430851508210401178664100
Faltings height: 3.6584
naive height: 69.6772
primes of bad reduction: 2, 3, 5, 11, 17, 43, 151, 2161, 5381689
regulator: 23384.1088294824406200534432265512570511736257300628682
submitted by: The Wren
submitted at: 2026-10-20 20:24:36 UTC
last updated: 2026-10-20 20:24:36 UTC

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

(-1119, 1354737)
(-15824, 1278577)
(7976, 165247)
(64299/4, -10229619/8)
(8336, 90207)
(8556, 237)
(-904, -1333623)
(14232, -908439)

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 20:24:36 UTC · history