curve #503

y² + xy = x³ − 28281535047551361775692143582418533847135576891434053609851017150x + 1829049510690519672186241686758536608498299114804907087658552721973708921009494938889647718482500

a-invariants: [1, 0, 0, -28281535047551361775692143582418533847135576891434053609851017150, 1829049510690519672186241686758536608498299114804907087658552721973708921009494938889647718482500]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 97738017931768289086255432500097975267778215327728790840160340302536473908163965331725668131447123194256756930
discriminant (Δ): 2512095595368693863767133985305778719914034779438783097016598135794906694759729380841296111645067350845676820928492464260516919725625166352196867800176072915301241336378162902592524656240640000
Faltings height: 35.8374
naive height: 456.8288
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 31, 37, 61, 83, 107, 109, 113, 157, 179, 383, 461, 1867, 2011, 3061, 9281, 9467, 16843, 19867, 61099, 178091, 179899, 116426291, 2532811447, 450736066760495349997
regulator: 75492338031106424251761637339.423093065181166841457512596757418014343963072
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:13 UTC
last updated: 2026-09-02 17:33:22 UTC

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

(-31136252503193694592207466902152582798594758839700/982351936281882721, 1597967314346888255324932064096406038858856884147898153375787203000580695250/973645045559164210778250769)
(62619692287140881182887398894799906597659129116780/629850083210640601, 60502283357312414271097212101820336546586829428527629741786803953273530/499868519266189667069560349)
(-9452291825044058498675890183734703392381983483449460/165318344722646549281, 3837380273420809413192863998317355368754440346816898324136882206297961271578590/2125600151121722090574073993679)
(84698114360033710138693893570196304793316400589028/220357274143864729, 714745919187873438375786064323149543774255003282473994943941891707335133066/103440613382126206667167933)
(31361247399521855884326104849541240170413229877000094/331210432685042662249, 55419363292390290210554927797362839101448579989968568895226489092856551693774/6027760830314833363981956501557)
(171048052440749667385162462949422680014990963041670568711392380/103726527193639262873849530009, 2225852898033243080493287028001614263471153916577474310711104013971898581827400680236890246250/33406790740628599435127884559819477564614973)
(607598686344129716582624150942031851355141360067948300/6111240721109996850361, 992746728086885806377001570925414855014952299191897211858908446871837100824250/477742729293245791827692277268141)
(138176780798550901641807231649529800969002279096222460/1374667568049002180409, 2200357405734268547073129386067425886201920087711184833898843829152892914592970/50967875356434863718952335988077)
(161090903770665321365106580546082351803083543793334520220/2424900907632541184892529, 1863063953297776438910612931634509932468480059426495192617531512930700805123396562690/3776078325627562015405208504248817417)
(91784274195268844822932235925170116297277483229334975/921501311673173446084, 451889673092658681917298631591648889132413247660035627715474056545306966325475/27973315049342757001460492954552)
(2585527244153107575555475962203245400773723726887060006048550/27551029832442028833595557481, 174186597043490307395942281777820395932004086981666872166134123254304025235457021955375250/4573058484083731268430890494663233030359371)
(56655208922195077260316966910639143906746426414809465392300/542696251443871313083537449, 1513515742999697583704146132257308908242246547053081116742645146749261743996117289662550/12642566092159699072309523803786300945707)
(454420640168288291544920367118169558716750382780164300/539101447162205946001, 300646774467834379103732067252421646153925891581970812986824000580496590285814050/12517158353055874508558367112249)
(-787180577410754106226867852028597116707340679291926400/5704039616316929761129, 758952400505915074714140040278130685954998900764738570090184385801093551436022650/430798120370413660797520709095883)
(353822343867763683237280435519519428580802379466637692/3844372238535109426201, 18045406520608680800948597954725543132725611283420376354058982031191579298038806/238362619021825403281228150898051)
(122436478591661674217624893936946231994330016935999013887820/470291325485085505483342129, 35493676826597393646131646057542708179482522498102760707607625481513753656437527375642110/10198832339594080327922966395477346151383)
(76089962656519923875940247304320395232732634860744134212614/17959574427475163914116169, 20972677787942811571676176004920647601446242567387525660924980374863502211691360253471154/76110410100686322437778388054988357947)
(194676796369138457607244411447302490884789553548213046257338617540/1908999645919325445356151886774129, 6173268094892991417947560505131421550003147576973125159616265497477978435915537832475489078808570/83408204379978674684311529284207694590094335978633)

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=241164/7043 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 13:41:13 UTC · history