curve #3001

y² + xy = x³ − 75922402743330943165726357898335x − 191001757341777787870494261104626287571932900775

a-invariants: [1, 0, 0, -75922402743330943165726357898335, -191001757341777787870494261104626287571932900775]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 35144361650725361785516703913690
discriminant (Δ): 12248415091634378937162944077391173844353929821283451628334995048984987729745327078481330176000000
Faltings height: 17.3392
naive height: 231.8354
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 37, 41, 43, 47, 139, 181, 797, 11813, 1518463
regulator: 6882814.63256508112751295670282577474931471315137581466
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:47 UTC
last updated: 2026-09-26 20:32:47 UTC

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

(10309376425153766, -349284367786641534886003)
(-242479138437557650/49, 86449092291088231524860795/343)
(-5984382020317730, 221425520055532005206365)
(-4790707216745050, 250538063077903474495565)
(-358337474696303530/121, 118054560800669350424393815/1331)
(2887429204915648797046105/7354944, -4905229549640693087927814674988614855/19946608128)
(75773783216838926379157356230/2675164461649, -19754498810680871631472151695536624133763195/4375480267321872857)
(2862439829748093241670/271441, -60175960205894603889505717546235/141420761)

Commentary

2-isogenous image of D-z6old678-115, rational kernel x=22456707641669839/4. 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 20:32:47 UTC · history