curve #484

y² + xy = x³ − 33846394113618634461087891162302595498863180539675028640x + 74293935017700922643822833966414204824696459059596203865321768280569673544384537600

a-invariants: [1, 0, 0, -33846394113618634461087891162302595498863180539675028640, 74293935017700922643822833966414204824696459059596203865321768280569673544384537600]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 45137010909088824799547895561717868348558143392087086144356474853426870259431025549767699410
discriminant (Δ): 97054307859641822030350672034695968444864497945208558276096460142227212066598782632200926616220498378460728516083657024809944976744775699798135517801596326626304000000
Faltings height: 30.8287
naive height: 395.1979
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 53, 67, 73, 101, 103, 373, 439, 523, 557, 823, 6871, 16573, 32257, 1025009, 13078867, 341910589, 1050385466742442173865787
regulator: 63571135768406371937.68441165023138172105475726412682389447714063831331
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:23 UTC
last updated: 2026-09-02 05:24:56 UTC

Witness: 16 independent points log in to add more points →

(7386595899278409033364630530, 476770657287089269189318166857113860101410)
(2755110347857587508882489440, 44231391140666971718822174737288202844000)
(40595183036697876184944393795/4, 7047723133855597014603939949486204845222855/8)
(-6586160985433357634655304470, 107333258767229056008153516914749194436410)
(4168624738593478792964697120, 75106677548545912486311854566315322309760)
(576196751029756617038780117176/81, 320667896135524524169951616601073102642049768/729)
(67849435203014859286788612816/25, 6156623939943514087205074902739414433547024/125)
(5062393861651124120685055680, 180799155557292293625818776297367662097760)
(195575672428164891807536161021320/26569, 2049718056565933246819269494738201235384942551520/4330747)
(260407906448336515639344060121695/58564, 1533686216451852556626637025033839755462110607405/14172488)
(315851764190557744432080252218940/109561, 944557743011590032769800844973796147006461751900/36264691)
(13309872490000139715878271536252150144314560/4796440214657569, 13856423623883705472511073487227365811978535558119819572473669600/332183879906676938264753)
(-163080158700803596197500516058480/27889, 1252048318903855868796684886112299309951442078880/4657463)
(-427166496999306101990106971621068080/172843609, 858846874428028069120416076769903289679352787881648880/2272374927523)
(304054034292740224346426377422737626980/80787261361, 340729079732329548362621565252873892228728585906660252660/22962244083898391)
(110057300712275272938631092115552447440/23868087049, 470292719591583365073110781368773113439730232213427266960/3687452372461157)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=9150/16213 on branch u=19/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).

last edited by Matthias Breddin at 2026-09-02 03:27:00 UTC · history