curve #5405

y² + y = x³ − 585999205

a-invariants: [0, 0, 1, 0, -585999205]
c-invariants: [0, 506303312904]
rank (lower bound): ≥ 7
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 67831124536816281
discriminant (Δ): -148346669362017206547
Faltings height: 2.5494
naive height: 53.9008
primes of bad reduction: 3, 3541, 24517
regulator: 175461.1310102870017975786166921910074055116275183502
submitted by: The Wren
submitted at: 2026-10-21 21:36:15 UTC
last updated: 2026-10-21 21:36:15 UTC

Witness: 7 independent points log in to add more points →

(921, 13972)
(4465/4, 226957/8)
(1353, 43483)
(1515, 53770)
(3025, 164604)
(14497/4, 1734709/8)
(7671, 671422)

Commentary

Rational 3-isogeny partner of https://elliptic-rank.icarm.cloud/curve/5380, submitted by Daniel Błażewicz. Completing the square gives the Mordell model Y^2 = x^3 + 86814697/4; applying the standard rational 3-isogeny and returning to a minimal integral model gives [0,0,1,0,-585999205]. The seven displayed points were recovered and checked on this endpoint; PARI/GP returned rank bounds [7,7] and a nonzero Neron-Tate height determinant. Same conductor 67831124536816281 as the source curve. No original-discovery claim.

last edited by The Wren at 2026-10-21 21:36:15 UTC · history