curve #5817

y² = x³ − 938758128301442700

a-invariants: [0, 0, 0, 0, -938758128301442700]
c-invariants: [0, 811087022852446492800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 312919376100480900
discriminant (Δ): -380707267731276075952348325397581280000
Faltings height: 6.0818
naive height: 96.2898
primes of bad reduction: 2, 3, 5, 17, 23, 103, 463
regulator: 156841.787469175225035887652987669693732565847305
submitted by: jesper-petersen
submitted at: 2026-10-25 11:00:22 UTC
last updated: 2026-10-25 11:00:22 UTC

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

(1001469, 256232997)
(3631269, 6851537847)
(134530669, 1560385367603)
(8544008929/8464, 233436763358767/778688)
(1393314681721/232324, 1641068594034337331/111980168)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-18,-4,-1,22) and specialization t=1/22. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=372927980. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:00:23 UTC · history