curve #28

y² = x³ − x² − 3289744045996x + 2272951313488252420

a-invariants: [0, -1, 0, -3289744045996, 2272951313488252420]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 2756084633910119906617080
naive height: 98.0791
Faltings height: 6.0398
discriminant (Δ): 46747845361418708281172757979630982400
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 59, 11016191591
regulator: 403784.1579848618784679002566306921904067755443292495
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:08 UTC
last updated: 2026-07-01 22:58:19 UTC

(1438008, 718248070)
(870276, 262859618)
(928733, 136859730)
(1166620, 151132950)
(15828084, 62574630898)
(-99958904/49, 241216239846/343)
(269482005/4, 4422204477635/8)

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 22:32:00 UTC · history