Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 9x $ |
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This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map
of degree 48 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2}{3^2}\cdot\frac{3285294025173x^{2}y^{30}-8015656437259308x^{2}y^{29}z+763915672054931472x^{2}y^{28}z^{2}+85067640416396374560x^{2}y^{27}z^{3}+564597662827988827152x^{2}y^{26}z^{4}-70559386773422326751196x^{2}y^{25}z^{5}-1013084198624151142593354x^{2}y^{24}z^{6}+6876500924766264885116928x^{2}y^{23}z^{7}+211127914873854837447968574x^{2}y^{22}z^{8}+767792685557411706436990056x^{2}y^{21}z^{9}-12579062171078785762977405489x^{2}y^{20}z^{10}-125061765986880227962366452144x^{2}y^{19}z^{11}-25254212315243069077529470533x^{2}y^{18}z^{12}+5244838576701579527579473309974x^{2}y^{17}z^{13}+25584270936247753165168595273925x^{2}y^{16}z^{14}-44260943636728155321192218625792x^{2}y^{15}z^{15}-826131226335798308454436142489877x^{2}y^{14}z^{16}-2231683565445086415504747703164612x^{2}y^{13}z^{17}+7024635906456153192709096531082556x^{2}y^{12}z^{18}+59277012894381430651389926133910176x^{2}y^{11}z^{19}+92189091859504736100490623655483308x^{2}y^{10}z^{20}-467192019058466534006184645306100830x^{2}y^{9}z^{21}-1739706222026222048645692672563036585x^{2}y^{8}z^{22}-333456019278080139044450294294278656x^{2}y^{7}z^{23}+3730024468112537048316787969088979339x^{2}y^{6}z^{24}+21133922088351275581176944505575752212x^{2}y^{5}z^{25}+36645923509398767937068164190461658697x^{2}y^{4}z^{26}-49216044730752946129913832911121000432x^{2}y^{3}z^{27}-111863729488668754805491115723403284451x^{2}y^{2}z^{28}-62763272438225288597968311389981618946x^{2}yz^{29}-11254292308472289566897534903256914055x^{2}z^{30}-104095383518xy^{31}+844286956869244xy^{30}z-265964386654815696xy^{29}z^{2}-14646213045522090684xy^{28}z^{3}+477837859974169244544xy^{27}z^{4}+29636129609607165989778xy^{26}z^{5}+126745074126960110915616xy^{25}z^{6}-7086567147275591544051756xy^{24}z^{7}-85913778583129260587105124xy^{23}z^{8}+228962632839193791139116096xy^{22}z^{9}+9780910984746037486453161888xy^{21}z^{10}+45697702779191477542520265723xy^{20}z^{11}-308088650944646775374913949806xy^{19}z^{12}-3865670861368023581161012441335xy^{18}z^{13}-7165883602990187201867347218528xy^{17}z^{14}+95161872132421390703560979285748xy^{16}z^{15}+617527071415526941856378624310702xy^{15}z^{16}+238298080252218329724482231110764xy^{14}z^{17}-11630437412116509828695055799280304xy^{13}z^{18}-45095773323078147279597113748795408xy^{12}z^{19}+24330557189698772093781326285901768xy^{11}z^{20}+584369296036974067954483917982813113xy^{10}z^{21}+1553597863532411917113384910144184304xy^{9}z^{22}+163518109016826865343729604089976054xy^{8}z^{23}-17641148748340235498277511461842373066xy^{7}z^{24}-36155785843728224549180860683284351640xy^{6}z^{25}+51951173109983906791761672082065882000xy^{5}z^{26}+124293472121750124950185514144193432693xy^{4}z^{27}+73223948829730004692469898960070903950xy^{3}z^{28}+13755264268382887991016017337586427889xy^{2}z^{29}+1532808577y^{32}-63710913435152y^{31}z+55160759925903636y^{30}z^{2}+94008758795283600y^{29}z^{3}-279036010305298138392y^{28}z^{4}-7018058135830545155520y^{27}z^{5}+92940146399671250754474y^{26}z^{6}+3120953341129949434245444y^{25}z^{7}+8521712797222703451841740y^{24}z^{8}-344249018538249266605963872y^{23}z^{9}-3104737416007857450488025072y^{22}z^{10}+6402833400640880940084285822y^{21}z^{11}+215013060092739797883556557678y^{20}z^{12}+799477177660285480320786467760y^{19}z^{13}-4458429360867850391143244294739y^{18}z^{14}-45163969141904763375559707032634y^{17}z^{15}-65824025297480598304681138446516y^{16}z^{16}+761995829854531028359057358346768y^{15}z^{17}+3962481691934467209379883419209492y^{14}z^{18}+1041301252263802719588778512468648y^{13}z^{19}-47786759424018279628311056822205552y^{12}z^{20}-162402496423269989657975323112543616y^{11}z^{21}-49261309468805389051973545167525507y^{10}z^{22}+1708408716008003707246971007227964650y^{9}z^{23}+3568204679710562262357393455668000998y^{8}z^{24}-5160177663542553286538376002646744816y^{7}z^{25}-12421010628539735806145695973390091792y^{6}z^{26}-7350292152508435034385978685899702702y^{5}z^{27}-1379153574268575872784659186448577734y^{4}z^{28}+6971183744719621434545213951514768y^{3}z^{29}+3235261961108042802384597579313413y^{2}z^{30}+922590942661528874468644563498222yz^{31}+122266688105012453454076838847297z^{32}}{298x^{2}y^{30}-221940619x^{2}y^{28}z^{2}+9216025660008x^{2}y^{26}z^{4}-39905130783159684x^{2}y^{24}z^{6}+6517958895059034636x^{2}y^{22}z^{8}+132419319865149218031x^{2}y^{20}z^{10}-199384245775610365388706x^{2}y^{18}z^{12}-24723788174010618929129763x^{2}y^{16}z^{14}+1030288046985931730437243494x^{2}y^{14}z^{16}+64748163534161706594555901839x^{2}y^{12}z^{18}-2347936124900509574860620867984x^{2}y^{10}z^{20}-7129406531662370434462015889820x^{2}y^{8}z^{22}+714667397176047574550514920196510x^{2}y^{6}z^{24}-4238851250582161379472970220472126x^{2}y^{4}z^{26}-14146065121593731343378059666003526x^{2}y^{2}z^{28}+79022846541489991846244435655653895x^{2}z^{30}-41721xy^{30}z+9987265944xy^{28}z^{3}-192332348081586xy^{26}z^{5}+361757259019955160xy^{24}z^{7}+34934679320927089446xy^{22}z^{9}+10725473253405045078198xy^{20}z^{11}+1162268511261431154425811xy^{18}z^{13}-70727274002871215663951304xy^{16}z^{15}-7337525505123342296723232555xy^{14}z^{17}+265474939747982744954200701744xy^{12}z^{19}+4398211604801154429214694173503xy^{10}z^{21}-195418834799120629159449923619660xy^{8}z^{23}+806217312959973725051201474886795xy^{6}z^{25}+15717850135143998764461521574835818xy^{4}z^{27}-96583479106277362840680513580593777xy^{2}z^{29}-y^{32}+3643888y^{30}z^{2}-344822522292y^{28}z^{4}+3196516626143028y^{26}z^{6}-2433516818522985288y^{24}z^{8}-284178737165148666576y^{22}z^{10}-30656301419396199087585y^{20}z^{12}+3005772665917265036920602y^{18}z^{14}+371557066000234245887669832y^{16}z^{16}-13647608286090113901727132992y^{14}z^{18}-373633907789876507785999368648y^{12}z^{20}+14806668657789021574689146117082y^{10}z^{22}-49292534545020567860344119988920y^{8}z^{24}-1571785012962278891849962108272600y^{6}z^{26}+9755906979923195554320534596359593y^{4}z^{28}+293461728850716147398538y^{2}z^{30}-79766443076872509863361z^{32}}$ |
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
