Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x y - 19 y^{2} + y z - z^{2} $ |
| $=$ | $34 x^{2} - 23 x y + x z - x w - 19 y^{2} + y z + y w - z^{2} - 2 z w + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1444 x^{4} - 4 x^{3} y - 437 x^{3} z + x^{2} y^{2} - 29 x^{2} y z + 741 x^{2} z^{2} - x y z^{2} + \cdots + 34 z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^2}{2^{14}\cdot17^5\cdot19^3}\cdot\frac{1155718071727137998354098379963999925858750xz^{11}-5795342719886221373437140929075697097554000xz^{10}w-16908523370234697647420360021041485160848000xz^{9}w^{2}+30816225294535217704494693174250945492504800xz^{8}w^{3}+6485101040105861977236466356666993637423200xz^{7}w^{4}-13237620831026408732204014491380899297605120xz^{6}w^{5}+21415845456948561810718094119349166133670400xz^{5}w^{6}-6482450014306981600948443611847352721387520xz^{4}w^{7}+5169435213396314877267998134705966607278080xz^{3}w^{8}-1299783271460529868398030339213314061888000xz^{2}w^{9}+538416069638348942653286436853505681364480xzw^{10}-79973366007894582346456215592598931187200xw^{11}+372828342778648718583710243380152982567375y^{2}z^{10}+13491600171592275440007765245011927535103200y^{2}z^{9}w-34504278720269024339951198187207254928117100y^{2}z^{8}w^{2}-35922403051001593567202536725860991800173440y^{2}z^{7}w^{3}+61843418819740812050119410212899790932895040y^{2}z^{6}w^{4}-59267604579089083549329213736303579758666624y^{2}z^{5}w^{5}+11147052173074215842012215223226726144841920y^{2}z^{4}w^{6}-1346960167893346873552802249567850771508224y^{2}z^{3}w^{7}-7201878040480764808695873784929801173032704y^{2}z^{2}w^{8}+1072729416623143869206182303989812270161920y^{2}zw^{9}-954443803353774048491014486243145889506304y^{2}w^{10}-406166394413313150035994704776019765735125yz^{11}-123786412515437523476964586134990147097175yz^{10}w+3896951576240181606072854910362482283580025yz^{9}w^{2}+24451473615354750500721630571052567234078760yz^{8}w^{3}-49711408588984465858346288293744457021631360yz^{7}w^{4}+18883904951981092735386548622699922217270496yz^{6}w^{5}+2320668434111116415348947258729929007477200yz^{5}w^{6}-11857285580473701159969611934929227306120704yz^{4}w^{7}+5634650387824250647386367323619266830378496yz^{3}w^{8}-256189473726404304009792684180593541139200yz^{2}w^{9}-209665616101500550102284902379176168688384yzw^{10}+277244164079298986811421611718243334868480yw^{11}+46852568547561088912795062123462453735125z^{12}+566451891389178798841227095323739789238425z^{11}w-861824005844378540074961560382416665825400z^{10}w^{2}-3801185599194777815688010348085820317435760z^{9}w^{3}+2044266730124934050640329899949780446630560z^{8}w^{4}-2148603130210752652477867154273710349019696z^{7}w^{5}+625461679618283099335383975327835381393920z^{6}w^{6}-1549969811976803061973993931862239800893696z^{5}w^{7}+660991999206992346294783778146835794042624z^{4}w^{8}-604247824411644251241146114594710593166080z^{3}w^{9}+223432135078868524536467082976142827881984z^{2}w^{10}-65309958392550048819675333384444865704960zw^{11}+9377192406160578005677754127073202803200w^{12}}{447628987430566350104895xz^{11}-7129967036326687426689450xz^{10}w+46862274223623395097924915xz^{9}w^{2}-324200881937883435610623060xz^{8}w^{3}+748469557808619826735817325xz^{7}w^{4}-918804656099914899864436770xz^{6}w^{5}+675932240403076209696836025xz^{5}w^{6}-312738781038041761943984640xz^{4}w^{7}+79921031259016210506332160xz^{3}w^{8}-8121093335119779394406400xz^{2}w^{9}+1080196314670339832398804y^{2}z^{10}-9636997100817576108169980y^{2}z^{9}w+32541656939462732282273048y^{2}z^{8}w^{2}-55769414397003756130054472y^{2}z^{7}w^{3}-7602258421131121157648700y^{2}z^{6}w^{4}+270782029178569440521234996y^{2}z^{5}w^{5}-574923882052344626761287680y^{2}z^{4}w^{6}+592040727659019316530162432y^{2}z^{3}w^{7}-334631868917722907881259008y^{2}z^{2}w^{8}+95394175634227695295887360y^{2}zw^{9}-10985790590052285241036800y^{2}w^{10}-56852437614228412231516yz^{11}+14697882917020488037065yz^{10}w+5983040502673497226099183yz^{9}w^{2}+126456659465741505759738278yz^{8}w^{3}-240788831410773318143660310yz^{7}w^{4}-31340474264758887672642059yz^{6}w^{5}+563604822525857046812544575yz^{5}w^{6}-771721198764525114336558528yz^{4}w^{7}+535892983451365421723617792yz^{3}w^{8}-206857601852285431832052480yz^{2}w^{9}+42186093400109316471552000yzw^{10}-3582835294905785026944000yw^{11}+56852437614228412231516z^{12}-462326870347586838141960z^{11}w+1334954829301347542563517z^{10}w^{2}-14105334903623024878959143z^{9}w^{3}+49856686012174407155440770z^{8}w^{4}-73335419501100310056579466z^{7}w^{5}+52761148125567548488726945z^{6}w^{6}-19141503466781739491622747z^{5}w^{7}+4416299243014777085858048z^{4}w^{8}-2048068030427386959022080z^{3}w^{9}+238855686327052335129600z^{2}w^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
15.48.1.b.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 15w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 1444X^{4}-4X^{3}Y+X^{2}Y^{2}-437X^{3}Z-29X^{2}YZ+741X^{2}Z^{2}-XYZ^{2}-53XZ^{3}+34Z^{4} $ |
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.
