curve #3187

y² + xy = x³ − 57376946618861787090787658290x + 5289793363296814684546397379863278967957636

a-invariants: [1, 0, 0, -57376946618861787090787658290, 5289793363296814684546397379863278967957636]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 3953388363442151568066959453452856637833267252586
discriminant (Δ): 865990573175423503916881419417995780536176737460523462517246143427090471689767616512
Faltings height: 15.1872
naive height: 210.2719
primes of bad reduction: 2, 3, 17, 23, 29, 31, 61, 73, 109, 241, 277, 313, 449, 653, 953, 1039, 1399, 2441, 16139
regulator: 80465490717249.7168914257849924228263792381476513961736225123637355
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:42 UTC
last updated: 2026-09-26 20:41:42 UTC

Witness: 14 independent points log in to add more points →

(242560674141188, 2375631346522482750974)
(145429899457796, -145915003286519025154)
(202922314117268, 1415117272008187871918)
(144674017245188, -130191573412088897026)
(184875226768388, 1000519718859597784574)
(393909487538614244/2809, -5515167520590651512855834/148877)
(14122906844521940, -1678124107843147915777426)
(717284942759969252/1849, 511641729763087998497876186/79507)
(6202429803640303076/44521, 146298880193863391757108266/9393931)
(-19734507309107740/121, 4273600604034170236254106/1331)
(636160033502414599811776276/4644701694649, 233255442646300923131033981384481688550/10010061370134654893)
(1495384707898374087170538212/11372899651129, 5248866898360613498272959988976636175398/38353659715176859117)
(395786792715416/25, 261766825861766998239874/125)
(959997774638785183387076/2413462129, -798907597951239211859699044725386222/118566154011383)

Commentary

2-isogenous image of D-z2old141516171819-069, rational kernel x=-1100953370271777/4. Source: https://dujella.github.io/tors/z2old141516171819.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 20:41:43 UTC · history