curve #2986

y² = x³ + x² − 689158410651830771934365490246023060x + 220182537650149240889364112070994007271068052546304708

a-invariants: [0, 1, 0, -689158410651830771934365490246023060, 220182537650149240889364112070994007271068052546304708]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 6529827304278763417963592037876180
discriminant (Δ): 4227950808344993376606787623346355276101505737338850634872564481872972544516179407317849745117187500000000
Faltings height: 19.2939
naive height: 259.1759
primes of bad reduction: 2, 3, 5, 11, 17, 19, 23, 29, 37, 43, 97, 101, 227, 797, 4507, 16657, 216919
regulator: 83020859.1341720221566636681853789524173045860873064686
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:00 UTC
last updated: 2026-09-26 20:32:00 UTC

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

(456646453318362212425259224/134629609, -9489780032728690978471705245008224218750/1562107353227)
(27785968405176244022957464/1940449, -146225655570674369207536724214206156250/2703045457)
(66173212753899203166133936/136866601, 2851001296114495398675149781143140950/1601202365099)
(-859601160947373764, 421203562745187898324406250)
(763898559084675769591/1681, 2010574560782564798715655335000/68921)
(23241330272769834982131/15625, -3079539463892589748107292098587154/1953125)
(486175045421626236, -6800868607330534981406250)
(40614914756965860235286928935252634177784/123848975805155571794209, 7479455646409100549798858028203329109940308022475455374155590/43585158450853122007983246396560177)

Commentary

2-isogenous image of D-z6old678-075, rational kernel x=-241608309854211556. 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 20:32:00 UTC · history