curve #520

y² + xy = x³ − 82004726325843244842234221322638377298110833005x + 8481012781888088891168939580604853071753828770871517809118255345055441

a-invariants: [1, 0, 0, -82004726325843244842234221322638377298110833005, 8481012781888088891168939580604853071753828770871517809118255345055441]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 152600141322569268506676653888653486589666000303661800852107237946587629395343172143438
discriminant (Δ): 4220940463743691817552695733706774883678632857331130936686306134493474236265481798620611739985281683025401881296062354061307149651253900972032
Faltings height: 25.9234
naive height: 335.6829
primes of bad reduction: 2, 3, 11, 13, 17, 19, 29, 41, 83, 89, 109, 149, 277, 433, 563, 709, 827, 1091, 22109, 40531, 52711, 126001, 1354547, 2004787, 2350771, 21578390361659
regulator: 50838263571662937892965497.8835273402068593338854693274000085039398589032
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:44 UTC
last updated: 2026-09-02 13:42:44 UTC

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

(245476870488308107738680139714/1885129, 8429474408541374438936931176557872107850251/2588282117)
(-75000097556746912456526, 119203601498349699762418502026038655)
(-1139902671662317995457124515838/8048569, 2999349736911779194282491925977651423131995667/22833790253)
(31879702392084945903839262477364579/159912012100, 478165629889083709832067833481982154725453969231103/63947214518669000)
(-140332235587149272017976334535499836754301902/716530002437355646401, 2502943461589668954428348207010081448161908283155097108695026808111/19180131029693296176032813951649)
(1592363132855469348073634146606/7890481, 272121135006978790313553840612622102847797987/22164361129)
(5208127346571391435748013203075852470/139556786329, 11885286126730591397539646924947912815174651667545653527/52134647339283517)
(402566860049639694181285330/1849, 2447202809602741882475396125945981925677/79507)
(41413284326942771481253413432801235381468354204/203867031548089812784561, 1318797583504485669712576303790078707794192808190635645522388177473741/92049301859915190709770036814558441)
(484138088553303673277996005405939162/1219895329, 336863350396121122396778863299107903467475384037185569/42607284155983)
(505577127680788355907487186036594/223412809, 11280729557641509629558328147645275806601725182469/3339351256123)
(2619282797331396713941769057198386/10536817201, 63593089803395572753093360789579977780509596261447/1081593748865449)
(797401401912118680520112229056061604/13676838564841, 3157937960534410377018214503430055223584391013459084795/50579971594104847861)
(2571560684845442921658120523303030555179298/23873465263518016441, 3494697470976783657288805601846200827225540425410620571447658723/116646897705224447729393513261)
(12847158138409005694733772760796043749530/114239067281246209, 1007717508377293522799789167145040035518414560335322990563113/38611971138587267388412927)
(-14141543162859668839755449897447879139413790/439230920024096031841, 969311829106467648277685994914667431644985730661631415244063161039/9205329904679614020947695033361)
(40609666949051452717719717128373599536422254194/150704716996636936861225, 4512772611934484246203367073492670650599237833843128364159500221558827/58504634252118997510521691086599125)

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=-5088/9199 on branch u=38/17 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 13:42:45 UTC · history