curve #1709

y² + xy + y = x³ + x² − 83422626552087397705x + 293369117521705826265995362551

a-invariants: [1, 1, 1, -83422626552087397705, 293369117521705826265995362551]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/5ℤ
conductor (N): 827411177518827902
discriminant (Δ): -24047426620617204704265819905037365425866870236936476295168
Faltings height: 10.1693
naive height: 149.2256
primes of bad reduction: 2, 7, 11, 13, 19, 31, 47, 67, 11059, 20149
regulator: 9247.5976080670989241958386949742771241762044413597
submitted by: Michael Rubinstein
submitted at: 2026-09-25 06:13:48 UTC
last updated: 2026-09-25 06:13:48 UTC

(46619457415/9, 411111294999464/27)
(5236153487, 10797804388088)
(-142173553491/16, 36934363914190649/64)
(39718575067/9, 2852495849879588/27)
(499576493219/121, 184999564957007296/1331)
(7017444612891/1369, 1056370837070975948/50653)

Torsion Z/5. From Michael Rubinstein's github repository: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by Michael Rubinstein at 2026-09-25 06:13:48 UTC · history