curve #2967

y² + xy = x³ − 19465161172882545871432175062433690076x + 31673874725369875149701946844577835455707993010780190480

a-invariants: [1, 0, 0, -19465161172882545871432175062433690076, 31673874725369875149701946844577835455707993010780190480]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1315458511091655433538805332647246470
discriminant (Δ): 38615793529450755136912783555945908559130437471416139879628763515461325521751860657202183214614775809756784230400
Faltings height: 20.3640
naive height: 269.1987
primes of bad reduction: 2, 3, 5, 7, 11, 19, 31, 43, 47, 59, 61, 73, 79, 101, 103, 127, 131, 149, 421, 433, 4903
regulator: 1732690584.22923288969706676199434279845749855981721806
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:47 UTC
last updated: 2026-09-26 20:30:47 UTC

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

(8334917146496073273131784/2374681, -9396238970548725482036438899586865660/3659383421)
(5046724617429702011822364/982081, 7987015283578280216003184175158391596/973242271)
(-174317393926140171885762/96721, 234743899215161958475263391490123616/30080231)
(3298840421728417365166264/23409, -5988685078336988949220413227531803132/3581577)
(2080122959349423480, 429481163334355785761928420)
(2106940935186835008, 122728396591884791558535516)
(582229969978843184872345954602524/9959988843025, 14009931316951850663474813892881835478848000280528/31433176989200533625)
(229122318433481169685199108/74114881, -651775222437087268609846333069476213656/638055010529)

Commentary

2-isogenous image of D-z6old678-061, rational kernel x=-16875178086556225985/4. 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 20:30:47 UTC · history