curve #514

y² + xy + y = x³ − x² − 9688649196963872371542463851509033831935189067x + 370922517634930538957605782838566512600361610129476142250057405792091

a-invariants: [1, -1, 1, -9688649196963872371542463851509033831935189067, 370922517634930538957605782838566512600361610129476142250057405792091]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 78193173609204889224508498376195855635709965623422249377518655313399916501570
discriminant (Δ): -1229821661984755649934627616974482983435721592962363255355131867011699297303270922214609159302842459728454154044639338625455638114304000000
Faltings height: 25.3081
naive height: 329.2964
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 37, 47, 83, 131, 151, 199, 271, 313, 523, 619, 787, 1097, 1427, 2423, 3023, 4229, 5639, 5821, 15959, 27809, 140593, 175261
regulator: 16795087060466347431630591.3477608992943958968043482271203396013210727801
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:20 UTC
last updated: 2026-09-02 13:42:20 UTC

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

(4948100799489570488195534171/75076, 88855673559939611474089651586316074794161/20570824)
(1709633897326706447636794692491/30129121, 324866244380159356910787339599136172054669866/165378745169)
(179373141192472767882689018926163771/85362724561, 75887076588748343559124754327891340090954997396881306/24940341872262809)
(48692652566990898686460188721719572162931/832031087712779161, 1583145622120990001120548880078313141608910271627697408383074/758943297878669919729137341)
(15130936858205526007443702690791491/162616208049, 1087454137224438783992262445638333921941333781108818/65576124209215593)
(-644467921754817082454477574277977897301/8985711521898049, 22485669889459769876851616925952008000047652921846269843658/851782489395301105570657)
(213776206021076268303446513497409771/3063448573441, 31557118729179420447653335440275355293011988823391734/5361865198085152511)
(-5378495996768520579524963830672972173541/65455652417778529, 414361439867706361442402421101627957658914755333931585201238/16746371987001717803853167)
(309599263560942613309718875557331147811481971/4829845427982889243801, 1222199398582022543670185181863246086355844431552380353759859141986/335660199250385998619609141375549)
(748440433178611758685376771030677131/13331151904225, 96587311660057368248296890344671566514613840218604074/48674501865427756625)
(785915685308898904035588756333827339691/4481835335073121, 19127820762260694692005311228413423510723827930993369337334/300043238559661380830831)
(290168786589308433019102050606599179109771/3163273116031236961, 89672469894206138354195610885476850543363317476890111365732394/5626068758855146382896785041)
(5992197495574655249119020843878467488241831/180682599741282081, 14668215015489270419076582596505181297109023226788655847733687426/76802347762320753957526479)
(67835069115243942163050056035/514089, 13742202879883883887773222902607825622934866/368601813)
(2723694627292897774204961217726808459499/78953293482046369, 195607456190039898634445379849297679545764573881980657103542/22184792781941510387141903)
(90028927537565880432830307144243665867815331/2144151943870996282276, 613155498506454125098303607592393665125256787478766914095299863899/99284930730116690100779580709976)
(2033794526461466710263145240841495493886136411/10218529531833223441, 91719263386849835553276708585594035064103111294185772032861345848646/32664995792903596628222591239)

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=-5137/14798 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 13:42:20 UTC · history