curve #3020

y² + xy = x³ − 616132300130332020371996761870x + 185603159595723305641335231450041290536912900

a-invariants: [1, 0, 0, -616132300130332020371996761870, 185603159595723305641335231450041290536912900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 40671677445041477433310596010
discriminant (Δ): 87548021298889308131282306442204280873299720619317731961309524235585237521489332736000000
Faltings height: 15.9304
naive height: 217.3934
primes of bad reduction: 2, 5, 7, 19, 23, 29, 37, 53, 107, 131, 239, 263, 3329, 7971031
regulator: 5674590.34844152186113424041288787060845033807271773148
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:36 UTC
last updated: 2026-09-26 20:33:36 UTC

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

(16434083313668849140/32041, 12189501720225897921101387670/5735339)
(-711811809841060, 16233158065603405428130)
(11934929964754396/25, 64469800012437988787866/125)
(13519518900407255164/27889, 4273365559662189105238209486/4657463)
(8514695157228947100/29929, 29897598545591089995784966410/5177717)
(11364772098318535895826/28462225, 273620377318647192192557447347146/151845970375)
(-11546736210416251590/12769, 2625721021789947157763356710/1442897)
(91732129715656398740/212521, 27029038689789145006046935530/97972181)

Commentary

2-isogenous image of D-z6old678-066, rational kernel x=-236532652199492. 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 20:33:36 UTC · history