curve #3140

y² + xy = x³ − 219045904855545533248287021290864537240x + 958410865019162015568275571339857856218598670068189825600

a-invariants: [1, 0, 0, -219045904855545533248287021290864537240, 958410865019162015568275571339857856218598670068189825600]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4669857800272842393315326009855704719253530
discriminant (Δ): 275829981192542281284106419027755867172166346251235478954302611638302002084657762519350331279924599958856077731840000
Faltings height: 21.0543
naive height: 276.4606
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 37, 79, 89, 109, 113, 439, 4261, 5407, 8623, 673273, 907969
regulator: 136960623034.415051630882489844306463521332971096787326059165
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:09 UTC
last updated: 2026-09-26 20:39:09 UTC

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

(22946100749107332760, -89519962308669705395021397440)
(161210193274893466219990/1849, 63835569167590725724324068964169170/79507)
(157358382742855280216782673014/91145209, 62419301883746200225979830369632964534499506/870163310323)
(93578514918251210879804294416/1040772121, 28254879610656986582110210795260339410453192/33576349395581)
(16994519255154003772703896848935886265/1365006803078523136, -20242074679108022835294721287742214865532366120837319515/1594785299715127300495855616)
(1377416906644057351110912019960/7486921729, -1611468777868612351103239708409775095246882600/647820876445183)
(88681025525334772480, -823983943048634005448394875240)
(2020720755336739252998593603440/907452475609, 18972090142362962395117905870892924067284589560/864441950622560227)
(122119946063228656458210712982650153291974685526521712961228090234242137/1665800886854889197297269435384175649500355838116096, -41849865565604860265036695267222058856852649360890414659538867247320043086526566211520058857885601085322293/67988370660645068620355671860457273738835233256052106382841357974204404985856)
(29948659568239998365903129723952131979062381680/342769537144770775464923521, -5111698276434158190596859883141431757481841502375710500648571939553640/6346047625711861124320242653214567722431)
(1869383131380531799476867415/139570596, -34085539116048654194565345345655928425885/1648887021144)

Commentary

2-isogenous image of D-z4old9101112-016, rational kernel x=8316639696921356080. 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 20:39:09 UTC · history