curve #521

y² + xy + y = x³ − x² − 33566197191746945981131296884979692826624395499435530x + 3032120634830705526163872828259816574581803558316141714854218289443975930369097

a-invariants: [1, -1, 1, -33566197191746945981131296884979692826624395499435530, 3032120634830705526163872828259816574581803558316141714854218289443975930369097]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 1718736990744216764866876731339300090750683708547303359476484436171961457484390923649700042984950
discriminant (Δ): -1551306549751893115305894709001416873611627888531944030118687135671028022772991465911296252629409532242493188690418607766952717009704566107073773071220736000000
Faltings height: 29.2560
naive height: 374.9449
primes of bad reduction: 2, 3, 5, 17, 71, 73, 79, 83, 283, 353, 419, 433, 1429, 1733, 1951, 2389, 3461, 3821, 4289, 5527, 7517, 47419, 58897, 214943, 244691, 971951, 8225101, 671744669
regulator: 794868976475409632714869322.360572588480336726643683439767868882435921759
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:55 UTC
last updated: 2026-09-02 13:42:55 UTC

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

(4188876801643519298211477941904767489/183980087041, 119149886459935805060614248420040915914226683995563493415/78914394754409089)
(1149699461514600767280021062703996025756941/11744246114782129, 1052859076613629336858498279279911667500704103586245996577389995/1272734204185797524528617)
(1563524680190501365804925748095768152529/19261995544801, 77587552600272932446905098833670514650956949668039047327015/84537989884804324049)
(-9931128527017899839346334463127188179/188827618849, 176977717573737181022174952758123262477447363932134192545/82053719977501007)
(-253057916931506626364488817019488976631/1305134880625, 2237013523780004774974000882024467376042917759740481401247/1491018715998015625)
(-24509344036449958879301997200519249018219389423/880492818091465188289, 51892747242587239925188741517556775288292708084189026504729641062308695/26126950765171864202464526482337)
(-130656984429935528933292433787089646521868291828551/693725047115493977114281, 944689487807689760099299143502185273606832920142020746207358368746140338685/577804689972373496652960368724222571)
(-690179174567821957791037095430088822757061/21779512244799556, 6479604725763642290771065259089404081559000543665722005870316685/3214195022176241038197704)
(76659480125836169222286613293174895157609/47676276849601, 21094611364926106047421239626555782100350855059908005683034095/329195204067401834401)
(366016643876810986781713820018640588471241529/795324614142487081, 6541307517905793747030351369814117955553481103419335387045732307395/709278238149336340802456021)
(1257765509534084216162574368737522880523845179275329/11488871989435453766898481, 31863320232009821003149368337252607830059040159813980765834760365059467899855/38941805718993891933411654110880569129)
(10418425738360046010428292450682863083220367170629/171028598171681987283481, 77913434646968953036448497970331607332165841559303350854418093145920775895/70729909776541355508157522525514621)
(-859245345027025597764144939164548760525130732021/14846963896940055658801, 3955580064092034861638219297861446878901415375372552377340711601791748985/1809074611097983289278100390996951)
(-12332123856159337778077516236823466821113739519479679/681577126581244358464234225, 33918508442450333195821862030949280778782189867813270509888535864963459319272401/17793955942003743881997975126156239800375)
(201855189303224314419330209752894351004042752865/1048559318095470441649, 65351420720101369721054961203949035897398082834699594219369108417784399/33953887548235814753515715948393)
(-123757860914525539765179143342047553524121551706324128394871/11707496509685790463825498273510681, 2330903267927089034491334383315991269325313678231935076013738467383662434752619058830935355/1266764999750043228391349605085633882007443236408829)
(-27413326117047710040452499493757814750277769719687759/26288156034336581177943614689, 7464586210200377345163727898554635543028334819503419960560926052485920320593463655/4262262428401573639452343919976337864829137)

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=-1990/15339 on branch u=38/25 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 13:42:55 UTC · history