curve #2957

y² + xy = x³ − x² − 72559023274528225307745105124106004564x + 250187716595165127630868602951840938571779366299252125648

a-invariants: [1, -1, 0, -72559023274528225307745105124106004564, 250187716595165127630868602951840938571779366299252125648]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 15589465171966572283635649128662490
discriminant (Δ): -2591947211593298640086445789020175942365978164170711218040327927623437392486629702829194693418026874023437500000000
Faltings height: 20.7085
naive height: 273.2468
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 29, 31, 43, 73, 97, 109, 197, 379, 2609, 6451, 157177
regulator: 152581484.222777937762113774076100618112074853687919715
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:05 UTC
last updated: 2026-09-26 20:30:05 UTC

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

(144739503549353795688/49, 2692745695530420000485594385492/343)
(8136391552125116115872893617/2406589249, 778829292125264681660285574711248087616348/118060048788193)
(-8292715268854878063, -16781425775370101655920913156)
(285681729902787075909573/4624, -151318104186508588359765061038885267/314432)
(4420713064358286312, -3977157447235811945629143156)
(9931308566006705256084267/279841, -30472695997458223420779416600963338809/148035889)
(5460411138377800312, -4098014956896064516007953156)
(512191852505990685000167835768/92892677089, -119255196663200524617731801689751479504394172/28312108801216687)

Commentary

2-isogenous image of D-z6old678-100, rational kernel x=79135743506933709507/4. 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 20:30:05 UTC · history