curve #5738

y² + y = x³ + 1585387033308737972

a-invariants: [0, 0, 1, 0, 1585387033308737972]
c-invariants: [0, -1369774396778749608024]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 22159413951135999675678779986365429483
discriminant (Δ): -1085811283605663984108260219331906044667
Faltings height: 6.1691
naive height: 97.3379
primes of bad reduction: 3, 7, 347, 124322141842321
regulator: 16340839872692.077251456033825365535183291701680699881450101571
submitted by: jesper-petersen
submitted at: 2026-10-24 18:08:57 UTC
last updated: 2026-10-24 18:08:57 UTC

Witness: 12 independent points log in to add more points →

(-1133006, 361862252)
(-687470, 1122710291)
(566062, 1329198300)
(1380820, 2053812708)
(6865033/4, 20615642127/8)
(45509044, 307008373787)
(-96316806304775/110628324, 1119371629179509145737/1163588711832)
(35693833/64, 679025342565/512)
(319431280/81, 5782399087139/729)
(12213253465/53824, 15780689624214819/12487168)
(58383037510/229441, 139097461711123549/109902239)
(59302293736/34225, 16495738528200516/6331625)

Commentary

Constructed from the Burkhardt quartic using the rational Bruin–Nasserden parametrization with parameters (u,v,w)=(-14/6,2/6,-7/6) and specialization t=325/848. The specialization was selected through a three-stage Mestre–Nagao sieve over 328 primes. PARI/GP 2.17.4 returned matching computational rank bounds [12,12] for this model and its rationally 3-isogenous partner.

last edited by jesper-petersen at 2026-10-24 18:08:57 UTC · history