curve #5880

y² + y = x³ − 50263449923080455769

a-invariants: [0, 0, 1, 0, -50263449923080455769]
c-invariants: [0, 43427620733541513784200]
rank (lower bound): ≥ 7
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 16285357775078067669075
discriminant (Δ): -1091411020009447203572425725009073266301875
Faltings height: 6.7452
naive height: 104.2508
primes of bad reduction: 3, 5, 43, 149, 421, 607
regulator: 78232670725.70907971437780084937059725345737484497044
submitted by: jesper-petersen
submitted at: 2026-10-25 19:27:41 UTC
last updated: 2026-10-25 19:27:41 UTC

Witness: 7 independent points log in to add more points →

(8186453329/36, 740700703625075/216)
(676834842721/176400, -184835134934633231/74088000)
(495886750389852184201/39362649369156, 10902974072808488856878334366001/246959923812006192696)
(332347207944524281/29878505316, 188065436560478258173016489/5164619157891864)
(109903006144875157849/23814439040016, 805377518040089100313074001411/116214557773034240064)
(59814320208592061620801/11855432342083716, 11412583622115309400992363029579419/1290850937723510888262264)
(9581678214396058822640145061375561/175695290324957989711944324, -937767268626760222629052913333667595116233835458993/2328842831700468855748012802799951756168)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with b=-1 and c=-a-d. Parameters (a,d)=(141/8,-7/4) and t=d=-7/4 give w=(d+1)(a^2+ad+d^2)(d^2-d+1)=-5050737/4096 on w^2=D_{a,d}(t), where D_{a,d}(t)=L_{a,d}(t)^2-4(a^3+1)d^3(t-a)(t-d)^3, L_{a,d}(t)=(d^3+1)t^2+(3a^2d^2+ad^3-2a-3d)t-2a^2d^3+a^2+3ad+3d^2. Cubic normalization of k=(a^3+1)(t-a)w/8 gives u^3+v^3=73678033305. PARI/GP 2.17.4 ellrank bounds [7,7].

last edited by jesper-petersen at 2026-10-25 19:27:41 UTC · history