curve #5711

y² + y = x³ − 17949773085574903

a-invariants: [0, 0, 1, 0, -17949773085574903]
c-invariants: [0, 15508603945936715976]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 63643329150346501736733922108041
discriminant (Δ): -139187960851807799298237087650285667
Faltings height: 5.4223
naive height: 88.3759
primes of bad reduction: 3, 53, 514867, 97450543
regulator: 99887179000.0895949047576185139972943191842901381055329056604
submitted by: jesper-petersen
submitted at: 2026-10-24 14:43:36 UTC
last updated: 2026-10-24 14:43:36 UTC

Witness: 11 independent points log in to add more points →

(261969, 5345257)
(347667, 155156539)
(496869, 323599522)
(543687, 377838175)
(592125, 435495541)
(1538907, 1904348515)
(9122217/16, 26183612875/64)
(4386271/9, 8444153024/27)
(20292865/36, 86713001803/216)
(15934911/25, 61365703091/125)
(10581743188945/28579716, 27674077649987958301/152787161736)

Commentary

Constructed from the Burkhardt quartic via the rational Bruin–Nasserden parametrization, using (u,v,w)=(-9/4,1/2,-3/2) and specialization t=-617/824. The candidate was discovered by a three-stage Mestre–Nagao sieve and subsequently tested using PARI/GP 2.17.4. Eleven rational-point witnesses were verified exactly on the minimal Weierstrass model. PARI returned matching computational rank bounds [11,11].

last edited by jesper-petersen at 2026-10-24 14:43:36 UTC · history