curve #3138

y² + xy + y = x³ + x² − 12980816973425563651934842940321413834x + 2735969070118707315872873850038434985085925706214368471

a-invariants: [1, 1, 1, -12980816973425563651934842940321413834, 2735969070118707315872873850038434985085925706214368471]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 13234498274885365856717453272852852367581218
discriminant (Δ): 136752719644102341530239379866831459594304619659879481779262024903654763935704953706649387473658315894439310133504
Faltings height: 20.3988
naive height: 267.9832
primes of bad reduction: 2, 3, 7, 11, 31, 37, 43, 47, 71, 113, 149, 179, 199, 227, 233, 431, 677, 2887, 3019, 308213
regulator: 63509074844.0323403290569945165729752344852325417600676303054
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:01 UTC
last updated: 2026-09-26 20:39:01 UTC

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

(300240628426701938939/1444, -11762235011103428050885603511937/54872)
(2815768726944738953382931/804609, -295289899569186574021081004216347555/721734273)
(1795295884539673469829228429/116035984, 74006021642143296877431726984899013411947/1249939619648)
(-164222769537252572563988369729/126882002025, -188358241904249840242936330026836646740959909/45196003531315125)
(4640250599124424344130129394966203364953635/1272819012207948484518724, -2823412951424547977269117989754990765425727409775062094757092079/1435985255563042839669575533867527368)
(60753947701995513494291073/14641, -473545365724458272129304366749210403989/1771561)
(1755999665244205234146170758748087/422372249579809, -39425577507062359863186945126386755861515189216567/8680466494572611885873)
(3491442819818104275981844878186540446733921633295459858845910810657/22003354151020469298267923692959720889602954361, -6522221905621281128046153578348808456641461206604926539033419428291992814914853563292336812561280497/3263873614256712486684703976698701560592973123709972982356758905852109)
(3036364435415959774395974519/243204025, -160317966008087905277545188885701256519693/3792766769875)
(18763759968802292859468200532171951581715498637343354377065865645601/3276403209278511285496685779045574932995843167296, -63936532686550911163111408379409923934369789793046363724835765545421816443413828778516585865986058731/5930564232238350505738441539298129211170107868389360516460377194229510656)
(188861145970638683945069419563224345807058681056721297607004228609845746897598081/49474500433555650038182380345537357782725021065848975585292225, -1032932200576951021884788651897836161839518477884387205454800910585742896418210162633590388129208513470029527856897440099/347994296798410418231495828473052858699494499697213339086673068230153135542677198251568673375)

Commentary

2-isogenous image of D-z4old9101112-015, rational kernel x=1851990373201879945. 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 20:39:02 UTC · history