curve #5810

y² + y = x³ − 12334923204586401169

a-invariants: [0, 0, 1, 0, -12334923204586401169]
c-invariants: [0, 10657373648762650609800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 23648018451396414075
discriminant (Δ): -65728942760035030629403320054067310101875
Faltings height: 6.5111
naive height: 101.4411
primes of bad reduction: 3, 5, 7, 11, 13, 157, 397
regulator: 74444.6563842767513350853747404952837562611792728
submitted by: jesper-petersen
submitted at: 2026-10-25 10:58:49 UTC
last updated: 2026-10-25 10:58:49 UTC

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

(10533601/4, 19476628245/8)
(516146449/64, 11587561515001/512)
(9364371289/3364, 592959437177831/195112)
(10122790561/400, 1018087378489809/8000)
(292507566169/75076, 140739169635369441/20570824)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-12,-11,-1,23) and specialization t=1/23. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=36498927465. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:58:49 UTC · history