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}{3^8}\cdot\frac{334357107x^{2}y^{30}-307032930228x^{2}y^{29}z+34040257100568x^{2}y^{28}z^{2}-1012511883243360x^{2}y^{27}z^{3}-12834657540306192x^{2}y^{26}z^{4}+323560042784480124x^{2}y^{25}z^{5}+121459802279823769914x^{2}y^{24}z^{6}-9432785905029169938432x^{2}y^{23}z^{7}+355508909458121780113266x^{2}y^{22}z^{8}-8226892677799168880935464x^{2}y^{21}z^{9}+122457921443555676091282449x^{2}y^{20}z^{10}-1118629544263592428668584304x^{2}y^{19}z^{11}+4383713108207811119441362173x^{2}y^{18}z^{12}+32131942351786664595605159514x^{2}y^{17}z^{13}-650545487625779074843510223205x^{2}y^{16}z^{14}+5205214775641505828831051298048x^{2}y^{15}z^{15}-23754720614522808738863443855923x^{2}y^{14}z^{16}+44019609249555602192833681243428x^{2}y^{13}z^{17}+218982841872758126144652956417244x^{2}y^{12}z^{18}-2244480110556071470812482243783904x^{2}y^{11}z^{19}+10222485436034798822179660358065332x^{2}y^{10}z^{20}-29308957377433580995196042012553810x^{2}y^{9}z^{21}+49771281476864856700004705579097585x^{2}y^{8}z^{22}-10155801089874440891973860147137536x^{2}y^{7}z^{23}-227246363075475054560590264304152659x^{2}y^{6}z^{24}+799120845355014679817827880857351692x^{2}y^{5}z^{25}-1620963240311734424805027362674546017x^{2}y^{4}z^{26}+2228262722357959037427548931186197328x^{2}y^{3}z^{27}-2068232273203629640671161120808325269x^{2}y^{2}z^{28}+1179584468068929488187950976878729394x^{2}yz^{29}-312649692105646162082360172032146665x^{2}z^{30}+10422062xy^{31}-34807384484xy^{30}z+14206957524464xy^{29}z^{2}-1788334263164836xy^{28}z^{3}+128781518997798784xy^{27}z^{4}-6738486594951860338xy^{26}z^{5}+276300119317552726176xy^{25}z^{6}-8768830072520144634804xy^{24}z^{7}+205284986730508315024836xy^{23}z^{8}-3300790065795065490901776xy^{22}z^{9}+30880297530821697367714848xy^{21}z^{10}-27590506391185569638831043xy^{20}z^{11}-3762343916223624181866042306xy^{19}z^{12}+59599825525270058885216471655xy^{18}z^{13}-494455781284079436629544911328xy^{17}z^{14}+2252435431893643832310936414252xy^{16}z^{15}-1198991621043287180586860078718xy^{15}z^{16}-64179785237838244721399163765924xy^{14}z^{17}+532966076861219963020365398145936xy^{13}z^{18}-2403928209008682155509147157170032xy^{12}z^{19}+6311491548791396059389791583034488xy^{11}z^{20}-3760140761633407555407052612535913xy^{10}z^{21}-49272670617316467690350805482573136xy^{9}z^{22}+266303954961252197487113369369021946xy^{8}z^{23}-798278667904799153444217489225113286xy^{7}z^{24}+1634544221847791146629228761171402160xy^{6}z^{25}-2352055133380327995864132822154175280xy^{5}z^{26}+2298035903830164403595297307401087187xy^{4}z^{27}-1376181892528023189671011029613577310xy^{3}z^{28}+382127402689358852851793483082805791xy^{2}z^{29}+103823y^{32}+2647926448y^{31}z+587305642124y^{30}z^{2}-265263564129040y^{29}z^{3}+30374017809656632y^{28}z^{4}-1910357484411463360y^{27}z^{5}+78737836398245718246y^{26}z^{6}-2200456291917931436196y^{25}z^{7}+38890293412857711285780y^{24}z^{8}-285955832532279948859872y^{23}z^{9}-4858227076294559243925648y^{22}z^{10}+177443832234772844204196402y^{21}z^{11}-2622398689369245482420980518y^{20}z^{12}+22470945784719064092671491440y^{19}z^{13}-101269138249967565030294057381y^{18}z^{14}-73853057107717173348264013014y^{17}z^{15}+4779439346203003408237939234836y^{16}z^{16}-38249058062372278118421696779952y^{15}z^{17}+172686058063320600014137047286668y^{14}z^{18}-433373669439330259256169149932392y^{13}z^{19}-2618243937025641040666287559248y^{12}z^{20}+5317101712087755212815239807126144y^{11}z^{21}-26503651526987061981827602964228853y^{10}z^{22}+78662202630386320938921808129286310y^{9}z^{23}-161556424511668369845439176387078678y^{8}z^{24}+233829800442908188719692324931034704y^{7}z^{25}-229803496405087625125986472407683568y^{6}z^{26}+138346740814394862684847834956548478y^{5}z^{27}-38599150551545373873434835869394306y^{4}z^{28}-289733435234756126188387459632y^{3}z^{29}+329265149911892234638794435267y^{2}z^{30}-92370090585181784876839985598yz^{31}+8281591419570134591543729103z^{32}}{58x^{2}y^{30}+127381x^{2}y^{28}z^{2}-120428552x^{2}y^{26}z^{4}+60131108156x^{2}y^{24}z^{6}-19951845945844x^{2}y^{22}z^{8}+237269513407151x^{2}y^{20}z^{10}+5407006890367134x^{2}y^{18}z^{12}-196997925512070243x^{2}y^{16}z^{14}+3109055093137206774x^{2}y^{14}z^{16}-26914265693833250961x^{2}y^{12}z^{18}+58916572566107416416x^{2}y^{10}z^{20}+1379246208172163064900x^{2}y^{8}z^{22}-17041213470636384568290x^{2}y^{6}z^{24}+92335140891236465557794x^{2}y^{4}z^{26}-250997558305288457596086x^{2}y^{2}z^{28}+270933741526900654995975x^{2}z^{30}-921xy^{30}z+705144xy^{28}z^{3}+97643214xy^{26}z^{5}+10271266680xy^{24}z^{7}-79912856915994xy^{22}z^{9}+3576335343412518xy^{20}z^{11}-90498069943536429xy^{18}z^{13}+1773381437460440856xy^{16}z^{15}-28260037948588301835xy^{14}z^{17}+355734730552964138064xy^{12}z^{19}-3389714396031607747137xy^{10}z^{21}+23198189423954867950500xy^{8}z^{23}-106045720486696590779925xy^{6}z^{25}+286642766457643273459578xy^{4}z^{27}-342958490470193024234097xy^{2}z^{29}-y^{32}-2752y^{30}z^{2}-10414132y^{28}z^{4}+8207600788y^{26}z^{6}-2145137488328y^{24}z^{8}-239805720199856y^{22}z^{10}+9261919091912895y^{20}z^{12}-188826855118235478y^{18}z^{14}+3034038319903889352y^{16}z^{16}-37672789983809566032y^{14}z^{18}+345904070946325858872y^{12}z^{20}-2315900678812381196118y^{10}z^{22}+10622977445181364481160y^{8}z^{24}-28066629826938736245960y^{6}z^{26}+21813724220731982717193y^{4}z^{28}+57116712326649451507098y^{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.
