curve #513

y² + xy + y = x³ − x² − 854587859660320176976190174171721125177606330x + 8593155853329372722486227496348290854806889561122987902436023874297

a-invariants: [1, -1, 1, -854587859660320176976190174171721125177606330, 8593155853329372722486227496348290854806889561122987902436023874297]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 28977213133631261182797956136132989566312790364730373629599921717272285151559950
discriminant (Δ): 8043983691981932930289123818275852398131566994856457857097032952126234370293978013135704388682314250301287716342561728893350792256000000
Faltings height: 24.8095
naive height: 321.9912
primes of bad reduction: 2, 3, 5, 11, 29, 37, 53, 79, 83, 107, 151, 179, 193, 223, 227, 349, 467, 479, 521, 647, 761, 863, 1321, 1367, 1567, 2957, 334993, 9836003, 1166024137
regulator: 5994334272502409943.46890884001842830885301136751995834060434981185804074
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:13 UTC
last updated: 2026-09-02 13:42:13 UTC

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

(133665179871590627347839, -47775280780823544998811447350749895)
(11230897903166785077513, 641836784448154607364054604299415)
(2562893425126927154199249, -4102680312768652062532240810554966425)
(-20830643533980545338311, -4166056582268379948190523433484145)
(1120680038481368350580685, -1185976200675693487177417325258540129)
(-19109467362069526476501, -4236229120772000742708215701299425)
(-24692726000744577874353381/1156, 162537173098232871511026716661321703225/39304)
(194496920415822063016839, -84852887842176004925542289614583945)
(585454701088453215923917/9, 407930273364925394207010525077364917/27)
(-29633966617293892201311, -2809673449572184350313954850217545)
(1624818851228312885074687/49, 1402372017323576974829233915240845103/343)
(328899661320623196004569, 187899537405379437789477382195700215)
(21891309646065193697439, 613244173985668446149171568238855)
(55407365073189139347427689/961, -366055458437090236523892378625928341175/29791)
(6451144352471994784114791/2209, 256892847460951506385528611098287247225/103823)
(206819176004087011564400239/17161, -473442154494027045249311832235248579925/2248091)
(10214415042700802605905, 964237703524679874390966614591431)

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=-3688/7737 on branch u=29/13 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 13:42:13 UTC · history