curve #2948

y² + xy = x³ − 14912084074894346406365753643565006325090x + 699071143677082840447638868104731887058217501080642915290692

a-invariants: [1, 0, 0, -14912084074894346406365753643565006325090, 699071143677082840447638868104731887058217501080642915290692]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 194702283408067369042637179695724289610
discriminant (Δ): 1105648256292700046714660322063142704096721866323889318170417461479210467838014509010249740571649373913818824000000000000
Faltings height: 21.9032
naive height: 289.1226
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 107, 109, 317, 42569, 75217, 99881, 821039
regulator: 259382365.825469450149368411565441159771860512261759624
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:39 UTC
last updated: 2026-09-26 20:29:39 UTC

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

(514054065361445680324, 11352226819454827981453856409838)
(705260194295013500416/9, 2923141610654231562963690313126/27)
(75689751056131762924, 63254523617629215045094745638)
(147313922300455119077660284/1256641, 1056014134794145660395024383874312800918/1408694561)
(62008343726414001654, 113233128894864075200878584168)
(-973584631196637271691/16, 75214490969680849341937968549007/64)
(603564895249076687865063844/7789681, -2041233295025680874121194985458394733502/21740999671)
(13222703214285837537609916/208849, 8921294749637897990914958183333634814/95443993)

Commentary

2-isogenous image of D-z6old678-077, rational kernel x=-36711517139185885084. 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 20:29:39 UTC · history