$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}9&4\\8&27\end{bmatrix}$, $\begin{bmatrix}19&18\\8&11\end{bmatrix}$, $\begin{bmatrix}25&2\\0&25\end{bmatrix}$, $\begin{bmatrix}29&24\\32&43\end{bmatrix}$, $\begin{bmatrix}37&36\\40&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.r.2.1, 48.192.1-48.r.2.2, 48.192.1-48.r.2.3, 48.192.1-48.r.2.4, 48.192.1-48.r.2.5, 48.192.1-48.r.2.6, 48.192.1-48.r.2.7, 48.192.1-48.r.2.8, 48.192.1-48.r.2.9, 48.192.1-48.r.2.10, 48.192.1-48.r.2.11, 48.192.1-48.r.2.12, 240.192.1-48.r.2.1, 240.192.1-48.r.2.2, 240.192.1-48.r.2.3, 240.192.1-48.r.2.4, 240.192.1-48.r.2.5, 240.192.1-48.r.2.6, 240.192.1-48.r.2.7, 240.192.1-48.r.2.8, 240.192.1-48.r.2.9, 240.192.1-48.r.2.10, 240.192.1-48.r.2.11, 240.192.1-48.r.2.12 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} - y z + z^{2} $ |
| $=$ | $x^{2} - y^{2} + y z - y w - z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 2 x^{2} y z + x^{2} z^{2} + y^{2} z^{2} + y z^{3} + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{231yz^{23}-379590yz^{22}w+26372340yz^{21}w^{2}+8680536yz^{20}w^{3}-1332821712yz^{19}w^{4}-9074218464yz^{18}w^{5}-34285959360yz^{17}w^{6}-91199609472yz^{16}w^{7}-186923828736yz^{15}w^{8}-309169652736yz^{14}w^{9}-423693649920yz^{13}w^{10}-488702177280yz^{12}w^{11}-478590787584yz^{11}w^{12}-399367028736yz^{10}w^{13}-283738669056yz^{9}w^{14}-170733797376yz^{8}w^{15}-86111944704yz^{7}w^{16}-35771645952yz^{6}w^{17}-11893997568yz^{5}w^{18}-3013607424yz^{4}w^{19}-528482304yz^{3}w^{20}-50331648yz^{2}w^{21}-2209z^{24}+491676z^{23}w-7082478z^{22}w^{2}-601645784z^{21}w^{3}-3733013280z^{20}w^{4}-12478757568z^{19}w^{5}-28300511008z^{18}w^{6}-46447542144z^{17}w^{7}-53971559040z^{16}w^{8}-34645157888z^{15}w^{9}+19863604224z^{14}w^{10}+100638339072z^{13}w^{11}+182534922240z^{12}w^{12}+236705316864z^{11}w^{13}+246273884160z^{10}w^{14}+214007742464z^{9}w^{15}+157904437248z^{8}w^{16}+99425452032z^{7}w^{17}+53293875200z^{6}w^{18}+24096276480z^{5}w^{19}+9031385088z^{4}w^{20}+2726297600z^{3}w^{21}+629145600z^{2}w^{22}+100663296zw^{23}+8388608w^{24}}{z^{8}(3465yz^{15}-3466yz^{14}w-193380yz^{13}w^{2}-1175736yz^{12}w^{3}-4048144yz^{11}w^{4}-9600096yz^{10}w^{5}-16959168yz^{9}w^{6}-23188096yz^{8}w^{7}-25065216yz^{7}w^{8}-21621248yz^{6}w^{9}-14902272yz^{5}w^{10}-8140800yz^{4}w^{11}-3461120yz^{3}w^{12}-1105920yz^{2}w^{13}-245760yzw^{14}-32768yw^{15}-33135z^{16}-267852z^{15}w-1142338z^{14}w^{2}-3241544z^{13}w^{3}-6753888z^{12}w^{4}-10832320z^{11}w^{5}-13748128z^{10}w^{6}-13973376z^{9}w^{7}-11368576z^{8}w^{8}-7242752z^{7}w^{9}-3396096z^{6}w^{10}-927744z^{5}w^{11}+86016z^{4}w^{12}+245760z^{3}w^{13}+122880z^{2}w^{14}+32768zw^{15})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.