curve #3114

y² + xy + y = x³ + x² − 2475534555539215368217047352627195213840859838x + 44636253372774745448826509445383423713422239248108305481606377768906

a-invariants: [1, 1, 1, -2475534555539215368217047352627195213840859838, 44636253372774745448826509445383423713422239248108305481606377768906]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 326885688138289438101160708711925723710232867288025
discriminant (Δ): 110213147510288112580184857157721285003593504394456318327356453600820794646540484137734842402735522211005690062577763285038916521156640625
Faltings height: 25.0456
naive height: 325.1820
primes of bad reduction: 3, 5, 7, 19, 31, 37, 41, 53, 73, 83, 97, 109, 157, 271, 359, 431, 449, 457, 797, 1447, 3061, 6148501
regulator: 425342752621.707133941384786602167187116134779284551866583310
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:56 UTC
last updated: 2026-09-26 20:37:56 UTC

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

(81684668640388711098178235/418609, -1799674399066133568908337456990559638585534/270840023)
(6792676064113736821427851/81, -15060283156545262675557752588939974630/729)
(207273363996061859899213986729665/5158686976, 1173782897508637766423545802580910030812596673783/370517533364224)
(72221293508194178466866766482261159747351/66986330074980625, 19388432572463173526095314896623127857240019665754281661818824/17337212713624023542234375)
(208838586401405823952521459074802297115/765063353041, -3017978281414208140645214894394732295005799455568606304462/669184848574548839)
(8314834573027893593486818360/11449, -756457026591840727464792992606824950574094/1225043)
(12907595459961543560602525802233406687927662633015/360325690634962962316155841, -9489922314962608584184409708543669132034518227672485961775491068670944412/6839791160098524080363154147528649641889)
(1105033681467133041046497817635/30946969, -228966947338579900270689663505638205478905026/172157988547)
(69233983317977227800966410884000/520341721, -540135016639631541715072760142118434384621564878/11869514997731)
(1102999991384688738531681463877361979015982136953856863590/2322984808617199672668664423093921, -36438230852429376386337736032401579189869349429442211152856902464430864784804190142302/111961718169692370211989468255653479575266244185969)
(6672766567272000603103170762545657025479885191105629516049508552199193537244269188361647557083052365950/191008701625918496582570531380503378696740056311009049315438588113964248572329681, -2345256422619202616675545088328299198177630409345854432481124304515370587619329249633840302658445029690238790344925257665497781082267426500906026797830023/2639852907916014767048762195433338546420972038316540051273366982663302070684907933707541408377236346850171774965037709671)

Commentary

2-isogenous image of D-z4old9101112-019, rational kernel x=28537664503990119507945. 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 20:37:56 UTC · history