curve #3210

y² + xy + y = x³ − 2097708172520403409502132772036551x + 24677184469813106054456703134649298266936931229498

a-invariants: [1, 0, 1, -2097708172520403409502132772036551, 24677184469813106054456703134649298266936931229498]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 1101143210509046379320062209869918451783745022537012850
discriminant (Δ): 327693377652947914688669850046473878020134882226082363512262698305954117765026315172008391228680250000
Faltings height: 18.1826
naive height: 241.7921
primes of bad reduction: 2, 3, 5, 7, 11, 23, 79, 89, 97, 191, 241, 337, 347, 383, 1223, 1579, 2543, 4271, 12619, 31219
regulator: 5043984268414812.1876004866961760109240822634316868881025676776465368
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:48 UTC
last updated: 2026-09-26 20:42:48 UTC

Witness: 15 independent points log in to add more points →

(147940940627910696, 54334638079566033152126761)
(226829553240477914832/5041, -1655435331759255639209456747254/357911)
(179634219471368411232/841, -2354714242417186854979330292146/24389)
(2438744241800850969/64, 138323365469663976186813857/512)
(324554606471602578288/169, -5845327917726600122017923378083/2197)
(8323183823369418363927/22201, -753827738038546663592918609960536/3307949)
(-21469744512670262389088542314/51063443765161, 1844751616145252511017321806290500107261228504/364892679834707269909)
(2297387885068464267, -3481493678545318218840028484)
(80416251768496853930560055771007/9793265468543041, 2742159815455930487777278324472001166501508495104/969150648710639613845089)
(30651908072873732643827440900998822/803587765037663761, -288120481683419925376009205742164631426584970985576/720360637454151231703910009)
(5105378662418574712751691/3667225, -11529454825512891051407930420569286156/7022735875)
(91242833567260606890007999171683/1567389716214529, -620020892114025384833872199385625574495875147588/62053386762325729676417)
(11408892145574335366622/292681, -231517140380544101216882169149499/158340421)
(41190391122714760585312983/222695929, 256657366841082859159532405231282639512/3323291348467)
(-2860883902309404890326517277/135598488169, 385300467786176187543947630474904224083448/49932380487888053)

Commentary

2-isogenous image of D-z2old141516171819-039, rational kernel x=-102020260065396217/4. 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 20:42:48 UTC · history