curve #3051

y² + xy + y = x³ + x² − 1839926530538856584164329942171450x + 29873101006545500680178813099094522047280615853767

a-invariants: [1, 1, 1, -1839926530538856584164329942171450, 29873101006545500680178813099094522047280615853767]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 65784691229449019755350055710
discriminant (Δ): 13122765432207051703735070901124124545461382246320371005216153266367862989972859970779715993600000000
Faltings height: 18.0042
naive height: 241.3987
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 31, 37, 53, 61, 83, 89, 127, 433, 953, 61417
regulator: 9112957.89489753138398156108279028056564737610971624288
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:59 UTC
last updated: 2026-09-26 20:34:59 UTC

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

(26323375698764171202167333953/25155812020249, -667021836484603198293626629988119570752973537/126170409068844013693)
(19030865410291017, 1322946856454299639383371)
(30274902252940537, 1385122149841520915844731)
(21934709591986537, 261392959659972909942731)
(27970127422757481, 540311127564830747685579)
(-11536576828500983, -7040179091594253740552629)
(-7032549128278607/9, -151080217417016534407570583/27)
(27003268326679874557/961, 17843511340482205241180959721/29791)

Commentary

Voznyy (2021). Source: https://dujella.github.io/tors/z2z4old678.html (curve entry 30). Published independent points; exact point coordinates checked. One witness is a half of a published relation, giving a 2-saturated independent set.

last edited by Natanael Wildner Fraga at 2026-09-26 20:34:59 UTC · history