Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 9x $ |
![Copy content]()
![Toggle raw display]()
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{2^4}{3^2}\cdot\frac{458484x^{2}y^{30}-116893080x^{2}y^{29}z+9089621847x^{2}y^{28}z^{2}+23350628160x^{2}y^{27}z^{3}-26602864418844x^{2}y^{26}z^{4}+67561800397200x^{2}y^{25}z^{5}+13790571049341801x^{2}y^{24}z^{6}-1859774227182546408x^{2}y^{22}z^{8}+2260874704726371600x^{2}y^{21}z^{9}-57381736108371195609x^{2}y^{20}z^{10}-5176178157718713100680x^{2}y^{19}z^{11}-67257953267345317593144x^{2}y^{18}z^{12}-510344841388210930066680x^{2}y^{17}z^{13}-2279335037664841611389460x^{2}y^{16}z^{14}+4695508136345182843197600x^{2}y^{15}z^{15}+160605196274250867888725004x^{2}y^{14}z^{16}+1159457806490020752637582200x^{2}y^{13}z^{17}+3387548214517561851761751591x^{2}y^{12}z^{18}-7442911619680532703235665120x^{2}y^{11}z^{19}-108057513397641489842038370676x^{2}y^{10}z^{20}-426223591563172392116964263880x^{2}y^{9}z^{21}-560068702257005139067891173810x^{2}y^{8}z^{22}+2168421383145673619115079591800x^{2}y^{7}z^{23}+13000781301658221645539332472952x^{2}y^{6}z^{24}+31116619970281474925353288137720x^{2}y^{5}z^{25}+29395710543931319385304087573827x^{2}y^{4}z^{26}-46885995443084318122040269682760x^{2}y^{3}z^{27}-210414993836917209233922352748268x^{2}y^{2}z^{28}-390903803067749938979399597672280x^{2}yz^{29}-326683834148137302374577797643795x^{2}z^{30}+47320xy^{31}-6358046xy^{30}z-371491920xy^{29}z^{2}+92688349248xy^{28}z^{3}-4870385518680xy^{27}z^{4}+27339991865613xy^{26}z^{5}+8182086604693560xy^{25}z^{6}+95127025880908152xy^{24}z^{7}+183006679704205680xy^{23}z^{8}+17941029568612923096xy^{22}z^{9}+387484784241387409440xy^{21}z^{10}+3362085054765347897484xy^{20}z^{11}+23769357717304531453680xy^{19}z^{12}+120384551040198066711630xy^{18}z^{13}-638207297205666981158520xy^{17}z^{14}-16698470866999786080784296xy^{16}z^{15}-136448677658415775070132280xy^{15}z^{16}-498556946183479824989653866xy^{14}z^{17}+893016851209269918151752720xy^{13}z^{18}+19825933449098456857577869776xy^{12}z^{19}+100237955090743084287637729080xy^{11}z^{20}+187021909184770113060129589818xy^{10}z^{21}-592169835417027894024971109120xy^{9}z^{22}-5013861357704612298519987039708xy^{8}z^{23}-14954669166139899130634198736240xy^{7}z^{24}-15205063054007764140274729793250xy^{6}z^{25}+49490746884563586609469299918840xy^{5}z^{26}+233794498911620670559406780317044xy^{4}z^{27}+456054377368749423222903896264040xy^{3}z^{28}+399280267699112323626543851905989xy^{2}z^{29}+2197y^{32}+1838920y^{31}z-563976612y^{30}z^{2}+45707357160y^{29}z^{3}-437207772582y^{28}z^{4}-52729956062280y^{27}z^{5}+759269638287072y^{26}z^{6}+7588766778996240y^{25}z^{7}-727764820817447070y^{24}z^{8}-11845786579745788080y^{23}z^{9}-73254198976002362376y^{22}z^{10}-578525510813790057480y^{21}z^{11}-3445547741491164436872y^{20}z^{12}+40955233419924889905360y^{19}z^{13}+868343891244547484721708y^{18}z^{14}+7204759528236205297553640y^{17}z^{15}+28928392736667587352362214y^{16}z^{16}-47544943435161172052593320y^{15}z^{17}-1305189609799975442394590124y^{14}z^{18}-7323379665959619849547221480y^{13}z^{19}-15384852829250438125345547022y^{12}z^{20}+44981158694896070560668093480y^{11}z^{21}+425100194594012944157928567804y^{10}z^{22}+1333737142458880387044229932960y^{9}z^{23}+1376348716264979824689144403143y^{8}z^{24}-4920130094878124090011917190800y^{7}z^{25}-23379526551489665093083752953076y^{6}z^{26}-45846586443441125567274846435360y^{5}z^{27}-40331413886574918186916828636104y^{4}z^{28}-201212767493720707596614280y^{3}z^{29}+451501702316750812496212404y^{2}z^{30}-419394231821956351859360280yz^{31}+175246875439888904169804117z^{32}}{136x^{2}y^{30}-5363710x^{2}y^{28}z^{2}-2271162888x^{2}y^{26}z^{4}-1697116037931x^{2}y^{24}z^{6}+130572137624184x^{2}y^{22}z^{8}+80200298486311272x^{2}y^{20}z^{10}-12034156445934021720x^{2}y^{18}z^{12}+566945662834677805020x^{2}y^{16}z^{14}-2910897306724841728776x^{2}y^{14}z^{16}-617293169676992343958110x^{2}y^{12}z^{18}+22510614875214893149843200x^{2}y^{10}z^{20}-333346965211975777732554885x^{2}y^{8}z^{22}+1949523048857922198469942896x^{2}y^{6}z^{24}+2124695000958331358400520959x^{2}y^{4}z^{26}-62415349634492676856364540472x^{2}y^{2}z^{28}+148695418365105736174136457735x^{2}z^{30}-8052xy^{30}z+59880456xy^{28}z^{3}+25821826947xy^{26}z^{5}-2373761821824xy^{24}z^{7}-3268610589928248xy^{22}z^{9}+438846918616175976xy^{20}z^{11}-2186101369338666120xy^{18}z^{13}-2331447188559350235480xy^{16}z^{15}+151735737551717976456456xy^{14}z^{17}-4046266080685805998046040xy^{12}z^{19}+41576568824892846738356289xy^{10}z^{21}+207765099833405734002822696xy^{8}z^{23}-8583813610478916660967327200xy^{6}z^{25}+69350406670637167700681642376xy^{4}z^{27}-181738856485713392638390465137xy^{2}z^{29}-y^{32}+268888y^{30}z^{2}-263649960y^{28}z^{4}+73024892280y^{26}z^{6}+75744135047322y^{24}z^{8}-8432031873665016y^{22}z^{10}-921434868763068348y^{20}z^{12}+172404750080732296320y^{18}z^{14}-9637240865038243344945y^{16}z^{16}+236037149449452751209552y^{14}z^{18}-1650978205319786495791392y^{12}z^{20}-38695068606322040479381152y^{10}z^{22}+904851035876452327823835972y^{8}z^{24}-6934997938483167545071195848y^{6}z^{26}+18357360726222228403626086862y^{4}z^{28}+133928842696971127671816y^{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.
