$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}5&13\\0&47\end{bmatrix}$, $\begin{bmatrix}11&30\\8&11\end{bmatrix}$, $\begin{bmatrix}19&38\\40&7\end{bmatrix}$, $\begin{bmatrix}45&1\\20&45\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.dc.2.1, 48.192.1-48.dc.2.2, 48.192.1-48.dc.2.3, 48.192.1-48.dc.2.4, 48.192.1-48.dc.2.5, 48.192.1-48.dc.2.6, 48.192.1-48.dc.2.7, 48.192.1-48.dc.2.8, 240.192.1-48.dc.2.1, 240.192.1-48.dc.2.2, 240.192.1-48.dc.2.3, 240.192.1-48.dc.2.4, 240.192.1-48.dc.2.5, 240.192.1-48.dc.2.6, 240.192.1-48.dc.2.7, 240.192.1-48.dc.2.8 |
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 $ | $=$ | $ x^{2} + 3 y z $ |
| $=$ | $11 x^{2} + 6 y^{2} - 3 y z + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 18 x^{2} z^{2} + 6 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^4\cdot3^4}\cdot\frac{2422235760523644764160yz^{23}-1784061430721698332672yz^{21}w^{2}+487300855560381923328yz^{19}w^{4}-60395709292515164160yz^{17}w^{6}+3475874617250807808yz^{15}w^{8}-107790100775829504yz^{13}w^{10}+1944506844905472yz^{11}w^{12}-20851273580544yz^{9}w^{14}+129807442944yz^{7}w^{16}-434115072yz^{5}w^{18}+655488yz^{3}w^{20}-288yzw^{22}-415589953975640653824z^{24}+234730743802563133440z^{22}w^{2}-31415891817563947008z^{20}w^{4}-3725077708781125632z^{18}w^{6}+1111083893597601792z^{16}w^{8}-75389771733663744z^{14}w^{10}+2395719508230144z^{12}w^{12}-41611108368384z^{10}w^{14}+409786304256z^{8}w^{16}-2219215104z^{6}w^{18}+5937408z^{4}w^{20}-6048z^{2}w^{22}+w^{24}}{w^{2}z^{8}(1670162816581632yz^{13}-105321578379264yz^{11}w^{2}+2536708990464yz^{9}w^{4}-29066926464yz^{7}w^{6}+160000704yz^{5}w^{8}-371232yz^{3}w^{10}+240yzw^{12}-286554636582912z^{14}-31137317858304z^{12}w^{2}+2411546494464z^{10}w^{4}-56259288720z^{8}w^{6}+573268320z^{6}w^{8}-2559672z^{4}w^{10}+4056z^{2}w^{12}-w^{14})}$ |
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.