curve #3117

y² + xy = x³ − 21906390297814432263288170749935924256686088x + 9083357239267359891526501593710396419636972744091639970715721792

a-invariants: [1, 0, 0, -21906390297814432263288170749935924256686088, 9083357239267359891526501593710396419636972744091639970715721792]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 16491334853285711166878253103438065925189351213350
discriminant (Δ): 637166811450579847049863154899461031443294269288512139411311648743825665452320750695923645292037122182041251122502454119169600000000
Faltings height: 23.9816
naive height: 310.9997
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 31, 41, 73, 79, 83, 241, 281, 5417, 488399, 41465951, 58252159873
regulator: 32196768437815.2172461087908158507092076069603469131400937354
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:03 UTC
last updated: 2026-09-26 20:38:03 UTC

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

(-238334247394450944883211476/49729, -700290842102978476438008060670411339912/11089567)
(-1228477620706938396030817043728/275194921, -611073292405502585840489928234334338480312104/4565208544469)
(-1209125428830402227238176164/346921, -42419674520685138370034199709744636887124/204336469)
(1116704099066690167693880084/190969, -23741919745973908906641377249740968340760/83453453)
(242760437021659422408936251584688/52775413441, 910907793083856641996875705765881246407755829720/12124042954387489)
(91696119322820620104427494408065466037131382181788212911168/13430401194996114434714014511538621049, -20752727665425901354837545422438008604177754000956451904131875745602916677869629764232088/49219078597339703409616342674030209766593274719154145843)
(53952023172856295589893175341710976/4823755653025, -11426423953298269391202820924045371542814851391062264/10594438659517072625)
(27004258203840885986363000198309978/901176185809, -4383895569765943752207700435168851192288050076753442/855489256717041127)
(9789629880624926844213019804634535560875001914647708706186250673824948306364008/1329741357874761163035460262943889975713294815040237981649, -24090703120454982289848097929771228570106689684471173188750744795372339820390339411524344746382841586271078999617146528/48489841520990056830053798579581130548069767562551843997746146339481790317897826618393)
(61183763137551236211750448028547064435620891753204313136/13723625560521294546753714268236841, -9160229283544001469019920261349557185492302580507333021444563446608790710182993880/1607693626975598516941442741328583893482025110884139)
(190898288262681617311005544010361556084910676248335942/3026177696353887041618711228481, -83178724952951072476461431485682438679337341093015331828603282377547461495215914/5264312223497886441664541077036305140572524129)

Commentary

2-isogenous image of D-z4old9101112-008, rational kernel x=2437698274691638852112. 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 20:38:03 UTC · history