curve #27

y² = x³ − x² − 3225667994796x + 2205916672708538820

a-invariants: [0, -1, 0, -3225667994796, 2205916672708538820]
c-invariants: [154832063750224, -1905911076227795039168]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 2705139105695704399945080
discriminant (Δ): 45883723249375846665777287372882822400
Faltings height: 6.0366
naive height: 98.0201
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 291701, 2186969
regulator: 486718.9016642236194594447475021625676457117043360219
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:07 UTC
last updated: 2026-07-01 22:58:18 UTC

(1124958, 29035380)
(707724, 526795542)
(885716, 209112382)
(333788, 1080007450)
(1219101, 292124718)
(6320106, 15305815332)
(-1935671, 1094134272)

Found by Penney - Pomerance (1975). A curve of rank exactly 7, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:59 UTC · history