curve #507

y² + xy = x³ − 8489423401566737820199340108202184276626586020827061311555x + 300397187216364741269573095252831403479003263003057649505426359435866116343304926011841

a-invariants: [1, 0, 0, -8489423401566737820199340108202184276626586020827061311555, 300397187216364741269573095252831403479003263003057649505426359435866116343304926011841]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 4234555647786645799327350113311740532609808811288652763141321341722263630766433744772980729646355774
discriminant (Δ): 174444835767871731867585305526562595360481450229131311345771661442073342132614103498882080171116685157692129468670537641042631632916407111879665233542154442450528333497106432
Faltings height: 32.1179
naive height: 411.7721
primes of bad reduction: 2, 3, 11, 17, 19, 31, 41, 43, 47, 53, 61, 79, 101, 109, 163, 191, 257, 557, 739, 3359, 24373, 29147, 32189, 36341, 281777, 51119741, 217991747, 714716037637249332961
regulator: 19679648114471751314244490.0848763993086144339581803595679834742414174223
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:29 UTC
last updated: 2026-09-02 17:33:27 UTC

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

(68100879821237655644088574078, 6171990469542346036871853558090242587114393)
(20117501624741157907686199148114398/323761, 659109289185302164419200624979115085923813719028097/184220009)
(2729658504974695036121678883166/81, 5286078240980427390182420491855361844161566769/729)
(117707198774883445219266459719364924/2002225, 6006251115394166518493315681440266867688338823562013/2833148375)
(-193199429173237784209989930020786600791634/2131643800225, 56133519433138693176243245214372140461728598887071764907814791/3112231922985503375)
(1883555812702124264108730301218, 2582006282651006485770604707596588979546110733)
(1856193076117758720351139086358330341445054/26569767140569, 951613159415471089611700163019000894761086696423557640919227259/136956176295804140003)
(390364594205967601930247611841433364399294/6878633971225, 21267308845328034838296064450407121170042532993178577172765723/18040696495841375875)
(93116988098286053302307781388030/1849, 61001565884921114376180961661929795272270944683/79507)
(16198034721898380982610500137152109395842007086/124298216485921, 2061547661577718547275083051086409879197557582769755471799700965158343/1385789753060265982031)
(3954500475807916340708978449562547436573699/49889444056516, 3947665034685631724974238032548868160782605587307141283969488061/352381416174410410936)
(227048889403184035990991755597386132766/2593763041, 1994733381695419054077343368260101797682390461878609439817/132097757915089)
(553848397453909874567380680065723649343927474/6998967657587889, 6523779269568144022384855804603809079936185508418218791736151382361/585532465406203976562537)
(-2336982434153565297720773626903547937894614/158544940480369, 41025574775412477779504440424404160840397696595460572014432822967/1996312751895768489847)
(-185400553674261013301529960715386739968877458/2825601053642689, 3601446653755294475591299353088989836706641108201201825529914910423/150198731812634321633887)
(-3483600326430932153384881834893062156218647339266/39005714789698491961, 4532861499219121601626652564325463461658565158851733596984675128085956395/243608457173930948440186486291)
(-826192788904714723695497162035470001762549970/7784089462335169, 1647857500129163940766536967949041806629990615557324570088008549783/686770651845013112590303)

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 3). Fiber t=13497/11680 on branch u=1/17 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 13:41:30 UTC · history