curve #5733

y² + y = x³ − 648619028293807258

a-invariants: [0, 0, 1, 0, -648619028293807258]
c-invariants: [0, 560406840445849470696]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 34611639706896014723792957766081
discriminant (Δ): -181745270149594783666445411921240371947
Faltings height: 6.0202
naive height: 95.5504
primes of bad reduction: 3, 7, 5189, 9949, 5426611
regulator: 12518243420326925.4635140006626449879822952992968549519393735
submitted by: jesper-petersen
submitted at: 2026-10-24 17:23:37 UTC
last updated: 2026-10-24 17:23:37 UTC

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

(3934177446645838/4299031489, -96733182583680527330085/281874597639263)
(17842021672964194833/20605807367044, -2209945792641639668289556195/93537218941279585928)
(23235877347751511128/2225100272329, 111973407150921272959997773930/3319130898926905733)
(3345567629221414393930/842287570440481, 192506511645481197142398921949878/24445030746249593713871)
(723320124292454940630681/6833178512693764, 615170786951062985501168838685004791/564851322388302146670088)
(6717532586850706033/5248002885904, 14469974664187833175570647017/12022397907178950208)
(1551087731753503720/603267550209, -1894549335717685192941557789/468559716049980927)
(29653430940166268193/33934213294864, 27007493558697946949597243473/197677244180277618112)
(2081976720021988291761/668246799220900, -93973610338910584462829074878491/17274493835855898823000)
(611094462488646392305/312101547628816, 14439053963257953993810100645591/5513709532623524467136)
(147054543373960235946996/19031219205079225, 56352367751917912970685809842668794/2625426819383449805694125)

Commentary

The rationally 3-isogenous partner of the Mordell curve constructed from the Burkhardt quartic with parameters (u,v,w)=(-18/5,1/5,-10/5), specialized at t=-2182/1387. PARI/GP 2.17.4 independently returned matching computational rank bounds [11,11] for this model.

last edited by jesper-petersen at 2026-10-24 17:23:38 UTC · history