curve #5822

y² + y = x³ − 26947573739162827

a-invariants: [0, 0, 1, 0, -26947573739162827]
c-invariants: [0, 23282703710636682312]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 107790294956651307
discriminant (Δ): -313706187544730981108676457989822723
Faltings height: 5.4900
naive height: 89.1885
primes of bad reduction: 3, 7, 11, 13, 17, 47, 79
regulator: 17913.1135501476135875306333655021996690778236580
submitted by: jesper-petersen
submitted at: 2026-10-25 11:01:30 UTC
last updated: 2026-10-25 11:01:30 UTC

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

(309127, 50916092)
(319683, 75651439)
(342939, 115691431)
(486759, 297291676)
(3214029, 5759680153)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(24,-18,-1,-6) and specialization t=25/6. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=63184121. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:01:30 UTC · history