curve #3142

y² + xy = x³ − 615411322294732283803288785027250860840x + 5538719221187391635372324416383813743465156669756429313600

a-invariants: [1, 0, 0, -615411322294732283803288785027250860840, 5538719221187391635372324416383813743465156669756429313600]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 31000471439531693917957022221199714058221545530
discriminant (Δ): 1664184493908733386068254585085809176300454477056389508312508481164956558156678807087667633912450105451479449600000000
Faltings height: 21.2429
naive height: 279.5597
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 67, 71, 73, 103, 127, 293, 1091, 682327, 102838432973
regulator: 3299878871.07795810827014301302224954130762275056448073487855
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:16 UTC
last updated: 2026-09-26 20:39:16 UTC

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

(34671110493770918750004/1849, 1934676178040911990365369070753164/79507)
(-21305867139566753520, 94757618715859900542658542360)
(-140823980180769576788580/6889, 55955830766308882932242116542440820/571787)
(-319982428856419447650208320/32615521, 19206374758511903734988452128233827541160/186267240431)
(-4565286891241286448, 90846571013826078600309586776)
(434851104684132361680, -9053524518644420246943534089640)
(-448387402951542959379120/22801, 344913413265498395233597595965526760/3442951)
(-2148837525085444110975120/96721, 2731382068027181875632023524190237160/30080231)
(4536129681809404980, 53296071177645173186531232360)
(4008824717924459681136/841, 1270459687338992356169640908016408/24389)
(1042262140963440593822928/26569, -885050863905891924581801888926869816/4330747)

Commentary

2-isogenous image of D-z2z2old1011121314-042, rational kernel x=-136324006169006490241/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:39:17 UTC · history