curve #4045

y² + xy = x³ + x² − 45452141x + 114932096265

a-invariants: [1, 1, 0, -45452141, 114932096265]
c-invariants: [2181702793, -99317693943845]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 26676134520303198
discriminant (Δ): 301228991741756295465444 ☆ record for ℤ/2ℤ torsion, rank ≥ 8
Faltings height: 3.2830 ☆ record for ℤ/2ℤ torsion, rank ≥ 8
naive height: 64.5101 ☆ record for ℤ/2ℤ torsion, rank ≥ 8
primes of bad reduction: 2, 3, 7, 29, 73, 127, 457, 673, 7681
regulator: 20790.5488616135501395074021144322586615712885975128218
submitted by: Michael Rubinstein
submitted at: 2026-10-07 22:22:14 UTC
last updated: 2026-10-07 22:22:14 UTC

(-860, 392075)
(4907, 97958)
(8708, 611723)
(-30961/4, 475531/8)
(4592, 53123)
(18193, 2295258)
(-309, 359247)
(5152, -135037)

Torsion Z/2. From Michael Rubinstein's github repository: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by Michael Rubinstein at 2026-10-07 22:22:14 UTC · history