curve #517

y² + xy = x³ − 4676250172766005158346023958175760666167598982071315x + 123028530148436527748552999736184956122566447737600037487539252762948104949281

a-invariants: [1, 0, 0, -4676250172766005158346023958175760666167598982071315, 123028530148436527748552999736184956122566447737600037487539252762948104949281]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 16502786707526974416379816040011713123070308567840902601861766573584920924211702
discriminant (Δ): 5690170417883235106130043656047492632420738141033822025180127320113421585670542023422479853407921889782144369545376498760077066396959303669428059954163712
Faltings height: 28.4553
naive height: 368.5366
primes of bad reduction: 2, 3, 7, 11, 13, 19, 23, 43, 79, 293, 397, 641, 709, 761, 997, 37589, 51169, 121843, 297457, 764459, 5968009, 9274813, 18096011
regulator: 3587994529051353905493328.09935679242622696341833954454287177077473762185
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:32 UTC
last updated: 2026-09-02 13:42:32 UTC

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

(-21759388581446599346177479083428787/2638260496, 54379759108214908998006549389965754284286269047579777/135511612116544)
(242884560082604191251594677731468087873144826/5970177811478562361, 159353183614397221825289498143871046392317886313943710563457548963/14587501007699830592322319891)
(61302708085872462553303757920643916526974/908515990276801, 9274679296339698780497655289510075401694103742376899162716679/27384124664011234744799)
(3159075331767571239696072838861359438/48999806881, 3245307320840728988818042919008137900629104054830792471/10846548251371279)
(16585615723118317487276365502715922686/449464317241, 8073204696584738357499944839621459160000392605144169221/301330317029028461)
(44382449276073154064796876, 53941845463136057357924318122521461481)
(993208226272286906633106785496989080175574/43733466803481361, 1545120863981906323235186910900959520103362789787282224890166681/9145782281075572093659641)
(24622352339622117148012956636503137470174/610176897694081, 91885411565849227222374588601188275750598389282549594793399/15072442674206844588479)
(354697055847627337115648187091569774/9339482881, 13097350331207266964267309943676973331521528409332601/902576965102721)
(43151829798668508577761758233532261846910/1061125608455929, 369854648503852656237560992819440205398588444477372008347707/34566084988780036068467)
(3249322786421981947967994313281531242478/80925155297281, 273170001962305658356458392200077364174972246944826992153/727989829954647608321)
(87489901326724262240504762366391778588857626370/92127961767303452809, 25813052479496367098808780134944019235675208393670188737633003126049473/884274689547131985662441982373)
(12184682662779026408862451273708729470/684949898689, 120880847284064066854551090600379738440763064280636901497/566876180303294113)
(562646020201283875413371374565915435405235/6859014256935556, 306652098900573601384209588975583175838081575061893628251496681/568057840341512463666296)
(44051081463650394831049730520052415305208460174/1152124410212886661489, 443360090740945209498148350227817882342600565807278706214839301659167/39106510677469550078639603514487)
(123972536396726656873006947463764/3478225, 262306681831455999499157194310814073030353310683/6486889625)
(8610051128604955445968251584382445868175067902943903458/223679570653952287367897596921, 827480349650422197473435477466147538959008294783149324539179472108168438226506231/105788748816096133649858681981590781077055219)

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=-20341/26177 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 13:42:32 UTC · history