curve #4651

y² = x³ + x² − 73785055x + 216933153009

a-invariants: [0, 1, 0, -73785055, 216933153009]
c-invariants: [3541682656, -187451494295680]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 148797332771970176
discriminant (Δ): 5374473654865220558358272
Faltings height: 3.4759
naive height: 65.9636
primes of bad reduction: 2, 7, 17, 41, 79, 103, 167, 271, 647
regulator: 134355.976748964336635454046476177865521971302039903510
submitted by: The Wren
submitted at: 2026-10-20 17:32:25 UTC
last updated: 2026-10-20 17:32:25 UTC

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

(-8103, 531834)
(178880, -75570247)
(70184, 18459557)
(80813/4, -21125197/8)
(136825, 50513878)
(2249, 249742)
(13525/4, -625649/8)
(-8920, 406747)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Eight 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 17:32:25 UTC · history