curve #518

y² + xy = x³ − 701429501154544521286381511411723243942581176537015x + 7056932368933554465437431007351574774882649922846633445924603245047947055017

a-invariants: [1, 0, 0, -701429501154544521286381511411723243942581176537015, 7056932368933554465437431007351574774882649922846633445924603245047947055017]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 132399771304839820510988825724471544529303829934050389830244032069096333604610
discriminant (Δ): 573035091826912912426444929148610619989680402756691849065020235490250642686808109945781778071288280003979030330566301364914248431896121963199910272000000
Faltings height: 28.1140
naive height: 362.8452
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 29, 47, 61, 67, 163, 173, 193, 229, 367, 397, 461, 593, 761, 1021, 1399, 1697, 3671, 4021, 6101, 6761, 7487, 7723, 43517, 122263
regulator: 362658138559063799071887.178764969600814616968748220327214177197924360989
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:36 UTC
last updated: 2026-09-02 13:42:36 UTC

Witness: 17 independent points log in to add more points →

(-635911889648194668547866, 86618257751572382574991932645786322773)
(222388385639007938555705104596/16129, 5353469910789300734649643097483406214105929/2048383)
(-19520952108308752460002947683376/3500641, 680506348142281681908942673585674920265655711683/6549699311)
(-11344135338019654957601403066/1681, 7385539168276530272460254785610883689804013/68921)
(86695319269884033308584154807697414/3664970521, 13498060594434793208815649000903916566275309898831487/221873650370819)
(54766057794814753361738608139616/3988009, 30035495100610414822141724947990410101538504849/7964053973)
(807185079441655471108903641667368294/33504007681, 394678962485402835459150523033220985896737807899269973/6132607069937921)
(143547174632379830198608674, 1692415976615822619090017779849102949613)
(-1418820961063097266925636424138/54289, 1098246284057754141915665922242050935991964877/12649337)
(3959820474045962728781736428576702262654/2206895935267321, 7898446042621360788214982452528123879869171571738116634736703/103674698764469386950131)
(54883795450668820190156626093195582672811498934/3148693315230784875841, 1986989922622262582242897138382522921103889721927700161052079736105493/176683316682895632330564861819361)
(629364033179767173664052750209514329466754054/35872557441012677641, 2640635783598702756864666153138417925653271554565860021685159751703/214854032669196911698379574811)
(60928563784495983740712089068915851123306/4432162319313169, 1034866818339089237399621675102852491408218079415253944475339/295068932709004418519303)
(-361156938779132463605738853748798169614961286/259862778566497094641, 375361200605550206988644148685670305158050434705257744590556878120667/4189055523370964248311378208439)
(66496302586194963312348153149967774311099470986/281249773656912504889, 17044004408884177386993671823838709190346370709028977454201016412233031/4716700196206303347502227072787)
(634512268928943752065103107362265965893046/4306882522146649, 497761467150198402915778816521721448749905567569683011927153951/282647103100362156899107)
(334977826557620944448175245391800355816/11229918125449, 4236812744286246283952227295439570625735324704754961915289/37632657239619022043)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=-8433/8669 on branch u=31/17 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:42:36 UTC · history