curve #506

y² + xy = x³ − 259496484824083234629242373329851001672356796420x + 50016117156480155928546279934867282922014958083580023771361969090547216

a-invariants: [1, 0, 0, -259496484824083234629242373329851001672356796420, 50016117156480155928546279934867282922014958083580023771361969090547216]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 34910793879807212391975584054857751022998385247520531253870511025704633110375003514482
discriminant (Δ): 37645050375796614367579106018424369183462386908948174109240355949677070095593714986746572250495892735027630973844189684936703133474174326013952
Faltings height: 26.1502
naive height: 339.1388
primes of bad reduction: 2, 3, 11, 17, 23, 37, 67, 71, 73, 83, 109, 193, 367, 499, 509, 709, 883, 1987, 4937, 5051, 7333, 52057, 148501, 256471, 374639, 502259, 571721, 579707
regulator: 103290734092434552179294771161.53533281905698468523904499348229729746632024
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:25 UTC
last updated: 2026-09-02 13:41:25 UTC

Witness: 18 independent points log in to add more points →

(-448431790495226969035593574578081308024/3677030956729729, 63005875980450401924400686824193415803901639348712278812740/222969740949491503669183)
(-65354368552582825736196042088534703480/578021177969041, 3878953798596192491232620338554847958028570975639276952484/13896826200235319523911)
(-21408072632619786747973243244707401283423489038/41336087256673054839889, 1792598661854674588004100181875115759916118934992048618088685361496488/8404154890245820272071997002936537)
(-2006736763713110540023680470672149608106863675032/3924138041192953667861449, 1720477304300194215555601596040784765289870241950103366066337982140354188/7773496628315090312158225523496349957)
(125411075665586085860544232763683069904024619204040/310916623108978898686857961, 574255461950662016228043139347179666378777100853401274629630748258590180884/5482339336134957801090764172121840088459)
(24350651769714961291437835551785396424/71293384618681, 21108335942056408817222302581009551101512842342902236300/601968616056602389421)
(46438493957865390337395898558141030483865278/142944098918678387929, 1255393824441068802348263348819472050388144405792358713846205138/1709028664705431180774614578717)
(2999200034047029250875490271714353464877805735320/8255730754841509290052849, 1440964082437688745049973628216209274489496527052677872477776969704129052/23721015714357514196632969994001505993)
(341218998586537245019248821091730053016982318475148/840865634243314523561529289, 2618870547695556697843738661782496844691390694901038090303673702017638121828/24383155323049319891161296264269103630587)
(146010188847593760915909946609613973760746758848547080/3097466379926516479882181755729, 1061118710529640150461504040284782485958486218957034396943564638031082583905701212/5451423246810219217633120960996101203860777817)
(-454309579222431696741582696419634900555768/857163601614677809, 156043666809744866759957376512282218101379160503818206830073316/793588894167090059831655127)
(68128953126896301903608879217342524464522600/261364049324813336521, 39086323998571116287923974075356062663981981204443403564860695924/4225409214845493867968620781819)
(245966494353956829567706455637388418150275597261894594446/583737827208328128283566811702129, 1755077711526722312218132043328413947107434176121010750611914398282173790042457362088/14103495226150207682000091413704084706035458731383)
(42333091226845171188758081426059567476851372910838265672968/99708251863294005010525044103630801, 4028837798557604508221533527518456669030454519654752646407711913469065312646967210077604/31484489294868396178569591644230505744442435494654951)
(5087297468653958974279635755778073027131116308996494880909000/6654549756600704904585061772686685449, 9377718422473426918983246110544156776427884823503834379618543138547114495490172613108099628/17166352055655084160937768049349438460640813022387097957)
(13395468455626466030333984283950604565036566590196232/39143718464621788758499822449, 278098795583362164629607144534677072684519833102236147891281116936758061757596/7744495268074689567480637662795384452984457)
(118226143875729230825749564281197701468195684796584/358900932076304331164928409, 113789626041118831447630740737191045908814201650726223620467664363580293068/6799263610860566072926179133329589002173)
(403076531519895254046553278090385142115190408981407046984/1202779565490276232083656417873881, 1095343234055562606805637562114509008762485639046548522814337705078534543569277071180/41713733445190514937130934554939126141546465232029)

Commentary

Exact rank 18, unconditional: rational 2-torsion point, 18 independent points (Neron-Tate height matrix of rank 18; 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 18 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=-4087/15630 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 13:41:26 UTC · history