curve #5806

y² = x³ − 170042511333489777675

a-invariants: [0, 0, 0, 0, -170042511333489777675]
c-invariants: [0, 146916729792135167911200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 226723348444653036900
discriminant (Δ): -12491044845379199779072139566727441071230000
Faltings height: 6.9483
naive height: 106.6883
primes of bad reduction: 2, 3, 5, 7, 11, 17, 37, 43, 241
regulator: 893076.003489776480518125946584564869926202114368
submitted by: jesper-petersen
submitted at: 2026-10-25 10:57:53 UTC
last updated: 2026-10-25 10:57:53 UTC

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

(100432171/9, 942899915944/27)
(2663871169/36, 137460738658753/216)
(84502904619/1849, 24542584334342172/79507)
(537134185281/31684, 386732152839950721/5639752)
(713440368921/91204, 483876339879351069/27543608)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(16,-10,-1,-6) and specialization t=11/4. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=5019111790. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:57:53 UTC · history