curve #5727

y² + y = x³ − 13365023545061

a-invariants: [0, 0, 1, 0, -13365023545061]
c-invariants: [0, 11547380342932488]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 25721835027844063303543851441
discriminant (Δ): -77165505083532189910631554323
Faltings height: 4.2219
naive height: 73.9705
primes of bad reduction: 3, 53460094180243
regulator: 4441143431.457319986771048531575589798947847207215747021473
submitted by: jesper-petersen
submitted at: 2026-10-24 16:58:48 UTC
last updated: 2026-10-24 16:58:48 UTC

Witness: 10 independent points log in to add more points →

(100633/4, 12796439/8)
(29971, 3681949)
(32341, 4523464)
(83623, 23903857)
(117787, 40258981)
(177337, 74589556)
(911641/36, 366191855/216)
(5799361/36, 13943599165/216)
(20716439/25, 94290417262/125)
(507874663/8281, 11109001158562/753571)

Commentary

Constructed from the rational Bruin–Nasserden parametrization of the Burkhardt quartic with parameters (u,v,w)=(13/5,7/5,2/5) and specialization t=-518/947. The specialization was selected using a three-stage Mestre–Nagao sieve. PARI/GP 2.17.4 returned matching computational rank bounds [10,10].

last edited by jesper-petersen at 2026-10-24 16:58:48 UTC · history