curve #6244

y² = x³ − 81442237042673830803843474649915863195675

a-invariants: [0, 0, 0, 0, -81442237042673830803843474649915863195675]
c-invariants: [0, 70366092804870189814520762097527305801063200]
rank (lower bound): ≥ 9
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 325768948170695323215373898599663452782700
discriminant (Δ): -2865386004990511745895882069092661965384978964982953271515747410376675754082320830000
Faltings height: 14.8847
naive height: 201.9246
primes of bad reduction: 2, 3, 5, 7, 17, 23, 41, 71, 139, 173, 179, 241, 443
regulator: 12714612799275.922221711804877492500527111798841628569510
submitted by: The Wren
submitted at: 2026-10-31 21:38:50 UTC
last updated: 2026-10-31 21:38:50 UTC

Witness: 9 independent points log in to add more points →

(51839991328351, 240565505734184408126)
(103858170857599, 1019228121732074011318)
(152362235374771, 1858906372731237541844)
(253552470959299, 4027301321930632444168)
(15069209907797179/169, -1740352391629869202567208/2197)
(1752258941206771489/22500, -2110089534197705208243709487/3375000)
(7849245647531189689/128164, 17667571744251216592173902413/45882712)
(42176318646208498201/722500, -210497269878982998511955806051/614125000)
(94273687919030291690881/1441872784, 24366468605270559900205808810029121/54750793354048)

Commentary

Rational 3-isogeny partner of the Choudhry–Shamsi Zargar–Voznyy (2023) rank-9 curve y^2 = x^3 + 3016379149728660400142350912959846785025, listed in the rank-9 section of Dujella’s Z/3Z table: https://dujella.github.io/tors/z3old891011121314.html The nine source points were transported through the exact degree-3 isogeny (x, y) ↦ ((x^3 + 4B)/x^2, y(x^3 − 8B)/x^3) and checked exactly on this minimal target model. Rank ≥9 is preserved under the rational isogeny. This target is a cube-sum curve.

last edited by The Wren at 2026-10-31 21:38:50 UTC · history