curve #3135

y² + xy = x³ − 1338030238082603163014505436054497501770x + 18838529738538375647591054661737968540822087057943137668900

a-invariants: [1, 0, 0, -1338030238082603163014505436054497501770, 18838529738538375647591054661737968540822087057943137668900]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 14957195101499347727901912655348880567251830
discriminant (Δ): 513083891855401048192109684639722035745091861022437955616593459862597668421659939362812544692356596284631040000
Faltings height: 20.9245
naive height: 281.8897
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 43, 71, 103, 151, 797, 94253, 253424587069203922319
regulator: 7674389289586.16487771178770341448355826763434083093900741565
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:54 UTC
last updated: 2026-09-26 20:38:54 UTC

Witness: 11 independent points log in to add more points →

(24262392069982068340, 25634044426632283614183209830)
(1034936550973860953140/49, -5699165676049384182391171670/343)
(21481178271245279470, -2891448187328691878503439720)
(52848766841115555074272/2401, -841361155887056401969615521224666/117649)
(235255174478782127023540/10201, -16176992322851677525888526027798770/1030301)
(11172312903670604345920/529, 19453998675281668323292211030/12167)
(21547627698974103610, -3423660275451050517742153190)
(21203263576034947752, -671528149552281842876219214)
(784278431849658206056/25, 10994920297321104154545254568206/125)
(21121245430551532780, -17381267277437713138736570)
(110326051389368741146690/529, -36115779741719952775776442982298730/12167)

Commentary

2-isogenous image of D-z4old9101112-006, rational kernel x=-10559122671197498060. Source: https://dujella.github.io/tors/z4old9101112.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 20:38:54 UTC · history