curve #2975

y² = x³ + x² − 516760309017918635609982278524120305x + 142718223904020170076561686183516223710316694927306600

a-invariants: [0, 1, 0, -516760309017918635609982278524120305, 142718223904020170076561686183516223710316694927306600]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 665963354664280732450924737051124420
discriminant (Δ): 32575015906842150197896809944442432316557689780691259221102269739783892007353014625766107664197894531250000
Faltings height: 19.3223
naive height: 258.3123
primes of bad reduction: 2, 5, 11, 13, 17, 19, 29, 31, 37, 43, 59, 83, 107, 241, 331, 2791, 5569, 775811
regulator: 940008349.199408383241450826490995796143693898203051789
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:19 UTC
last updated: 2026-09-26 20:31:19 UTC

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

(1566608531261812280155528985/2200735744, -38015261196339997853077468461057898671875/103240915222528)
(302152447127281940, 119009127290696087163306250)
(388728511380880145, 24072130062236555589890795)
(401066653121200510580/2209, 24322087545845325904709740458110/103823)
(981581068436218661494740/1692601, -429670591446758365231318992954153050/2202073901)
(1248776912503350255582885/2172676, -604069463380503445413212187025720345/3202524424)
(8629255649438874891069/22201, 79807176248612274033025261922855/3307949)
(19565338224391758635/49, 2188330219781126555041213475/343)

Commentary

2-isogenous image of D-z6old678-058, rational kernel x=-800777005254988881. 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 20:31:19 UTC · history