curve #3129

y² = x³ − 88027440704943205209713903944072452x − 469435073596985955704179266146861379743440533656704

a-invariants: [0, 0, 0, -88027440704943205209713903944072452, -469435073596985955704179266146861379743440533656704]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3046788250100588932752869018478456421677120
discriminant (Δ): 43559821348587958727543495527097653641964829717039800108013721936865104442316735347698301265723393026560000
Faltings height: 19.1516
naive height: 253.0025
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 29, 61, 109, 131, 139, 151, 163, 179, 307, 563, 761, 827, 2213, 11423
regulator: 19015700335.9839829827244686497299412471723553937592342448876
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:39 UTC
last updated: 2026-09-26 20:38:39 UTC

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

(209617812197080975147342121/82882816, -3013885079862340766111330182111549268395/754565156864)
(-52091056290089092931808534404071/220980949623184, 278397794954158590696701233652691349320682910845/3284976395995068882752)
(20842880086667298533427678828722/22838831442001, 89955898806002004237603091670910026220836968000/109146752622491336999)
(766666300792694775867324624392808358/744757717180732369, -642652378214775636971216015947389406045460833614431880/642721041445774823999503847)
(561353793239557780401823089411602/393146768429041, 13008707730030179683640543959585769144786843114000/7795286210990466486089)
(6901180880049671525517638368/10396453369, -512336331101396170041257592488989209952800/1060053574863347)
(141366143590712356286129274105703130836054590892/41071715886486887072058600961, -52953671820371066843951153057392391682689289921914963381163866733269100/8323658822543745194166422708228964275070209)
(15761045093871346955770753545673762/35509569153549241, -1464503861053494484985822597362959572318813716646160/6691417139158493717119789)
(803727931968418435881106340491403623242596815246957/2617464067429145809632361566878161, -5104457127689959243067454883280311493363796813412010009052070993370687310465/133912492189384223151179135679766664513017030909159)
(60765211016062192424545250923195483258702835302/24680448730642015059545339761, -14869595070145130580850591286277064761749957416453927503435397553898360/3877301537879371519306749929919160360623991)
(344734670147113515642879301/108889225, -6372482987054005649605897433928224542701/1136259062875)

Commentary

2-isogenous image of D-z4old9101112-020, rational kernel x=587981882605635556. 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 20:38:39 UTC · history