curve #6436

y² + xy + y = x³ − 8087240104664018096475081407923x + 9072136266534775122552825366854145436656183006

a-invariants: [1, 0, 1, -8087240104664018096475081407923, 9072136266534775122552825366854145436656183006]
c-invariants: [388187525023872868630803907580281, -7838325734286046288166928652771284603476803487821]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 9034171088450771262709353690866719130
discriminant (Δ): -1703440437381973062997347812685748155367049187108159687921589810558345114160102263844031250000
Faltings height: 16.6685
naive height: 225.1662
primes of bad reduction: 2, 5, 7, 11, 19, 31, 43, 61, 103, 409, 877, 997, 2969, 6037, 20323, 29789
regulator: 54314613261753.6863147490305725636347249139828834732469124128
submitted by: Maksym Voznyy
submitted at: 2026-11-03 04:32:33 UTC
last updated: 2026-11-03 04:32:33 UTC

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

(316347638603050382305/322624, -8367528439075314200358819514211/183250432)
(57684670660830587167845022930/118706189286169, 93772685817462380854754830598649107555561481/1293332065639672077053)
(-1153631728725209460478601045/355219576009, 6966442212301212119505795709954767106104/211711932960092027)
(121888398234619246406768014849936055/8337372567088069129, -41804383163739030622701686403624270030294540311757084/24073753730714186542767676133)
(74257072441044209310374931255/29546595319684, -10918007381480652423257209586248421196489321/160605778154109285752)
(4784357699769557744437159439680/917732070692649, 9161807368305047626790240098110125357182630889/27801860698645683222357)
(15678066786437943938107595547766995/7265087743127094241, 800144347983392466444741077128450885008098765418072/19582191606119269167775297711)
(4275666580769515103764148270/1467899441761, 179867693991828080379124984887309859328868/1778461458754933009)
(3394200999060027, 143964060783220585041644)
(303772997486965885070875895757195/99239459656748401, -3564224887855155272375944467950552170140667635148/31262707436714328347551351)
(78916161055703051320492481273395/45941343669383761, -155318534924268721184995491803941062250788734632/9847036316040676482398759)

Commentary

Voznyy (2026), T = 1897/605 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-11-03 04:32:33 UTC · history