curve #3004

y² + xy + y = x³ − 2516075952544956511230269091041406468x + 1536115990806522164493558393159722779862584397118669058

a-invariants: [1, 0, 1, -2516075952544956511230269091041406468, 1536115990806522164493558393159722779862584397118669058]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 24971493095101000569198256701990
discriminant (Δ): 45648673656937852714764289378032721541822788395045649967436068606968512448237175626474375396826640036000000
Faltings height: 19.5726
naive height: 263.0609
primes of bad reduction: 2, 3, 5, 13, 17, 23, 29, 41, 53, 97, 103, 211, 281, 971, 22697, 199049
regulator: 50803823.7302572451427685133835350015062051096179421517
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:54 UTC
last updated: 2026-09-26 20:32:54 UTC

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

(179621414078996820740269049/145467721, 979960538951022401902535759553502955115/1754486182981)
(264982684761137637216721/91809, 119058874981790130394055219712687505/27818127)
(4517791691032554558019442/4897369, 102218407677512192174475330841895586/10837877597)
(-377715091563866441211439/1181569, 1951123570508293699510497788446119345/1284365503)
(481065332898252689, 661096867632681894737885775)
(63919824419817873761/49, -236086000977109153286878839495/343)
(2131803731923622369, -2420847731689879709777596785)
(151278835491836821775340039815044241/163541139364840000, -935029617107943703559170227730062893107108636574061/66136396549647898648000000)

Commentary

2-isogenous image of D-z6old678-116, rational kernel x=-7298086438714790153/4. 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 20:32:54 UTC · history