curve #5867

y² + y = x³ − 51400294064499090769

a-invariants: [0, 0, 1, 0, -51400294064499090769]
c-invariants: [0, 44409854071727214424200]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 205601176257996363075
discriminant (Δ): -1141339779324135559002098045518035033301875
Faltings height: 6.7489
naive height: 104.2955
primes of bad reduction: 3, 5, 7, 11, 19, 31, 43, 283
regulator: 42475201.803062335857750534830637224791133896657471
submitted by: jesper-petersen
submitted at: 2026-10-25 16:46:12 UTC
last updated: 2026-10-25 16:46:12 UTC

Witness: 6 independent points log in to add more points →

(3224788147249/202500, 5754006418198778443/91125000)
(3578987449/144, -213752919931021/1728)
(2147650849/576, 9115073604145/13824)
(2106273409/36, 96653232002269/216)
(3150129481/576, -146413729967941/13824)
(9809279881/900, 952049073068279/27000)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, restricted to the plane b=-1 and c=-a-d. For (a,b,c,d)=(-6,-1,0,6), the genus-one square condition w²=D_{a,d}(t) specializes at t=-8/71 and w=245644/5041. After rational cubic scaling, the resulting cubesum curve is u³+v³=2759503285. PARI/GP 2.17.4 ellrank bounds [6,6].

last edited by jesper-petersen at 2026-10-25 16:46:12 UTC · history