curve #3002

y² = x³ + x² − 520100732095595577867575959539220x + 3975718267931975544291553909228912485758457056100

a-invariants: [0, 1, 0, -520100732095595577867575959539220, 3975718267931975544291553909228912485758457056100]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 104277816820885318880966408571860
discriminant (Δ): 2175805658939839376625116874525631322365156442238175617012374480909681981677035361498629526796000000
Faltings height: 17.7874
naive height: 237.6083
primes of bad reduction: 2, 5, 7, 11, 13, 19, 23, 29, 47, 53, 67, 79, 131, 811, 2659, 7541, 14633
regulator: 489541700.017622535946236745813708784059983191784729167
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:51 UTC
last updated: 2026-09-26 20:32:51 UTC

Witness: 8 independent points log in to add more points →

(-38570270536592082054440/1661521, 4029433256525839696924818371650850/2141700569)
(-158166274038084468079016/24117921, 315701704678428541499868946596416210/118443110031)
(-9770694417296069492315/376996, 49970257579624036060680880580975/231475544)
(2947184641357387839013936/390181009, -5329423117836516821446742537938454090/7707245470777)
(9543403483341410694195/546121, -190734391984281169408265332741290/403583419)
(18623783253712540, -865474842494318711424770)
(-4911556896576238227485598920/189520044921, -18226565120166934993088752507352280940630/82505466835863219)
(116142359850766880, -38861546931445499922172330)

Commentary

2-isogenous image of D-z6old678-068, rational kernel x=12974179118756332. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:32:51 UTC · history