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{2^8}{3}\cdot\frac{848x^{2}y^{30}+795744x^{2}y^{29}z-307057932x^{2}y^{28}z^{2}+35378933760x^{2}y^{27}z^{3}-2366564970528x^{2}y^{26}z^{4}+108449703656568x^{2}y^{25}z^{5}-3542688093265596x^{2}y^{24}z^{6}+80199093960404424x^{2}y^{23}z^{7}-1109899054179526176x^{2}y^{22}z^{8}+3988156090234542552x^{2}y^{21}z^{9}+187478027837313159564x^{2}y^{20}z^{10}-4571764943007081354192x^{2}y^{19}z^{11}+54738919037206867404852x^{2}y^{18}z^{12}-381407843669657550404112x^{2}y^{17}z^{13}+1152087625886886074044215x^{2}y^{16}z^{14}+5866542504214537099889664x^{2}y^{15}z^{15}-92663599811998322643245892x^{2}y^{14}z^{16}+557546455465771865712166296x^{2}y^{13}z^{17}-1792603641617196486626289501x^{2}y^{12}z^{18}+1027600145944623284405751648x^{2}y^{11}z^{19}+21112894603569533450120257788x^{2}y^{10}z^{20}-123114942104483035201863948480x^{2}y^{9}z^{21}+377786951886887589352203678285x^{2}y^{8}z^{22}-660958017064785917029233657648x^{2}y^{7}z^{23}+228412222019464180556394068184x^{2}y^{6}z^{24}+2384618305947754550316394541184x^{2}y^{5}z^{25}-8142809118308985043007611950852x^{2}y^{4}z^{26}+15083766293961979118494379902464x^{2}y^{3}z^{27}-17786315172996554448116147594256x^{2}y^{2}z^{28}+12587996158330677224179204575648x^{2}yz^{29}-4077834638194392679336538365560x^{2}z^{30}+224xy^{31}-202104xy^{30}z+14211072xy^{29}z^{2}+1314670176xy^{28}z^{3}-188279681472xy^{27}z^{4}+9435335315742xy^{26}z^{5}-196040531088792xy^{25}z^{6}-2996581123053756xy^{24}z^{7}+336041248457836512xy^{23}z^{8}-11291553532948495626xy^{22}z^{9}+222745006614383878824xy^{21}z^{10}-2782635697721155673952xy^{20}z^{11}+20408470578681060849768xy^{19}z^{12}-39079770957868237401375xy^{18}z^{13}-943139033109754208567784xy^{17}z^{14}+12339345850666370199996408xy^{16}z^{15}-78199403322453271982666136xy^{15}z^{16}+253554183488396638712675286xy^{14}z^{17}+148624385372970527813045328xy^{13}z^{18}-6414509523929805370427951088xy^{12}z^{19}+36885976068446587737123205176xy^{11}z^{20}-115354715126341369630101301323xy^{10}z^{21}+159967627666002763075558112472xy^{9}z^{22}+349993837694684812958207108784xy^{8}z^{23}-2716922772090426119750742391632xy^{7}z^{24}+8319943840299240921186906412380xy^{6}z^{25}-15921752954900323481480813852880xy^{5}z^{26}+19762572527847319455545418071568xy^{4}z^{27}-14685995799915772764318242660640xy^{3}z^{28}+4984020207886708773260225247816xy^{2}z^{29}+8y^{32}-25056y^{31}z+12105936y^{30}z^{2}-1497281760y^{29}z^{3}+106401677112y^{28}z^{4}-5820628766400y^{27}z^{5}+266281186814964y^{26}z^{6}-9808802722906992y^{25}z^{7}+273228789736277775y^{24}z^{8}-5448907986933976416y^{23}z^{9}+72601512854354403228y^{22}z^{10}-536428318849380838296y^{21}z^{11}-284239051264860599268y^{20}z^{12}+56622001611690677639160y^{19}z^{13}-695201079435972435556704y^{18}z^{14}+4535007915117954960158832y^{17}z^{15}-14708330199471980024446824y^{16}z^{16}-19670462439045812616582216y^{15}z^{17}+503640188626389209429843412y^{14}z^{18}-2904664221524805446072264904y^{13}z^{19}+9143005581645331574234881242y^{12}z^{20}-11530135257956950044481198488y^{11}z^{21}-39298345588126449216386225352y^{10}z^{22}+270628846125071176916815541040y^{9}z^{23}-823288783136552578439856817668y^{8}z^{24}+1582864140343264585556736112272y^{7}z^{25}-1976255420971445986773795864912y^{6}z^{26}+1476368835340456282118277413376y^{5}z^{27}-503438828697139776309258750216y^{4}z^{28}+2741602191679173672340704y^{3}z^{29}+835085725051702325483088y^{2}z^{30}-1985298138802160245488096yz^{31}+638131544614980078906888z^{32}}{32x^{2}y^{30}-39744x^{2}y^{28}z^{2}-48409488x^{2}y^{26}z^{4}+47651466996x^{2}y^{24}z^{6}-9518995462176x^{2}y^{22}z^{8}-2058553344414654x^{2}y^{20}z^{10}+158286058469068176x^{2}y^{18}z^{12}-5651670947986274598x^{2}y^{16}z^{14}+129125357621667613296x^{2}y^{14}z^{16}-1984449100141818522036x^{2}y^{12}z^{18}+19311465946547405633184x^{2}y^{10}z^{20}-91246892659590511741455x^{2}y^{8}z^{22}-197008214505429166259760x^{2}y^{6}z^{24}+5012269106691471169622139x^{2}y^{4}z^{26}-23750048104929546640468704x^{2}y^{2}z^{28}+36293731599976991987829255x^{2}z^{30}+104xy^{30}z+1137456xy^{28}z^{3}-1366353036xy^{26}z^{5}+702643734000xy^{24}z^{7}-179947210812984xy^{22}z^{9}+8002896484571712xy^{20}z^{11}-200085343951935294xy^{18}z^{13}+5171871931671687984xy^{16}z^{15}-163532869883575745040xy^{14}z^{17}+4319928183376961933616xy^{12}z^{19}-78068599740022292300277xy^{10}z^{21}+913598694474846196353840xy^{8}z^{23}-6589217529285086151530820xy^{6}z^{25}+26393221837052195417236512xy^{4}z^{27}-44370822539686971319919217xy^{2}z^{29}-y^{32}-8208y^{30}z^{2}+11023128y^{28}z^{4}-5838205248y^{26}z^{6}+2856570249132y^{24}z^{8}-959100662466864y^{22}z^{10}+46496773633970700y^{20}z^{12}-1332885374150312592y^{18}z^{14}+30852161912345410422y^{16}z^{16}-598460361115924038528y^{14}z^{18}+8990375190796635108192y^{12}z^{20}-95850962829090857729232y^{10}z^{22}+664657569384838337562390y^{8}z^{24}-2639996528882681899374480y^{6}z^{26}+4481534326186646980767018y^{4}z^{28}+31512668869875559452192y^{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.
