curve #3015

y² + xy + y = x³ − 19268484626923945756059529879144x + 32556145266192197851380590072573693098763094866

a-invariants: [1, 0, 1, -19268484626923945756059529879144, 32556145266192197851380590072573693098763094866]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1537557927687420856077617546790
discriminant (Δ): -28513369178462474352150321685562836712122700206799609101906756059560127391752936436535750220
Faltings height: 16.6372
naive height: 227.7217
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 41, 43, 59, 61, 73, 83, 107, 127, 977, 182057
regulator: 111570165.889800783019938610386276437656687139321598360
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:22 UTC
last updated: 2026-09-26 20:33:22 UTC

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

(259057134963548410/121, 44475819732405952816793782/1331)
(2598202547437216, 5682628374000453876686)
(22819300985144236/9, 27318291706558021311074/27)
(2840280405216016, 27227475895795428708161)
(201876273133321066/121, 94610658998391756698783017/1331)
(842912232180876619/324, 34847792991917189440250239/5832)
(489190992598429616059/192721, -90169914895500053714264167897/84604519)
(181818553298018630648491/51509329, -34127265819396741182733781106694661/369682454233)

Commentary

2-isogenous image of D-z6old678-025, rational kernel x=2534338701081673. 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 20:33:22 UTC · history