curve #6457

y² = x³ − 42031234691222436x

a-invariants: [0, 0, 0, -42031234691222436, 0]
c-invariants: [2017499265178676928, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 672499755059558976
discriminant (Δ): 4752218684337266390160101855832733307054794676342784
Faltings height: 8.6053
naive height: 126.4452
primes of bad reduction: 2, 3, 11, 17, 19, 59, 163
regulator: 22279.7946860881768229550972828595454132508145010
submitted by: David Renshaw
submitted at: 2026-11-03 18:31:19 UTC
last updated: 2026-11-03 18:31:19 UTC

Witness: 5 independent points log in to add more points →

(-203128563, 395491312161)
(-95856387, 1774314226377)
(-39138594, 1259003485080)
(611272332, 14237687237796)
(943474603938/1849, 839198807097608952/79507)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x, n = 205015206. Rank 5 example of Rogers, obtained at m/n = 18 in Family II, as recorded in Watkins, Donnelly, Elkies, Fisher, Granville and Rogers, Ranks of quadratic twists of elliptic curves (2014), section 12; https://www.dpmms.cam.ac.uk/~taf1000/papers/rankcongr.pdf. Witness points independently recomputed with PARI/GP ellrank (rank bounds [5,5]).

last edited by David Renshaw at 2026-11-03 18:31:20 UTC · history