curve #5402

y² + y = x³ − 413846530

a-invariants: [0, 0, 1, 0, -413846530]
c-invariants: [0, 357563401704]
rank (lower bound): ≥ 7
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 33830903740467681
discriminant (Δ): -73988186480402818347
Faltings height: 2.4914
naive height: 53.2052
primes of bad reduction: 3, 2441, 25117
regulator: 131230.3941731175606686103542689212090672792243467362
submitted by: The Wren
submitted at: 2026-10-21 21:30:24 UTC
last updated: 2026-10-21 21:30:24 UTC

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

(3417/4, 115799/8)
(946, 20802)
(996, 23962)
(1320, 43429)
(1698, 66946)
(1968, 84901)
(438801/100, 289957549/1000)

Commentary

Rational 3-isogeny partner of https://elliptic-rank.icarm.cloud/curve/5376, submitted by Daniel Błażewicz. Completing the square gives the Mordell model Y^2 = x^3 + 61310597/4; applying the standard rational 3-isogeny and returning to a minimal integral model gives [0,0,1,0,-413846530]. 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 33830903740467681 as the source curve. No original-discovery claim.

last edited by The Wren at 2026-10-21 21:30:24 UTC · history