curve #5846

y² = x³ − 78882275771489883

a-invariants: [0, 0, 0, 0, -78882275771489883]
c-invariants: [0, 68154286266567258912]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 25557857349962722092
discriminant (Δ): -2688082602144212080617533094824793648
Faltings height: 5.6690
naive height: 91.3366
primes of bad reduction: 2, 3, 7, 11, 19, 23, 61, 79
regulator: 112763.91568067279554497888872276313041062449507311
submitted by: jesper-petersen
submitted at: 2026-10-25 11:06:41 UTC
last updated: 2026-10-25 11:06:41 UTC

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

(584563, 347665108)
(640927, 429421090)
(1559539, 1927217956)
(2946463, 5049877342)
(85243291/9, 786991699592/27)
(142926721/324, 486571909663/5832)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-13,-11,24,-1) and specialization t=5/6. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=2918781558. PARI/GP 2.17.4 returned matching computational rank bounds [6,6].

last edited by jesper-petersen at 2026-10-25 11:06:41 UTC · history