$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&0\\28&43\end{bmatrix}$, $\begin{bmatrix}5&4\\44&23\end{bmatrix}$, $\begin{bmatrix}43&24\\4&17\end{bmatrix}$, $\begin{bmatrix}45&20\\20&27\end{bmatrix}$, $\begin{bmatrix}47&22\\16&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.l.2.1, 48.192.1-48.l.2.2, 48.192.1-48.l.2.3, 48.192.1-48.l.2.4, 48.192.1-48.l.2.5, 48.192.1-48.l.2.6, 48.192.1-48.l.2.7, 48.192.1-48.l.2.8, 48.192.1-48.l.2.9, 48.192.1-48.l.2.10, 48.192.1-48.l.2.11, 48.192.1-48.l.2.12, 96.192.1-48.l.2.1, 96.192.1-48.l.2.2, 96.192.1-48.l.2.3, 96.192.1-48.l.2.4, 96.192.1-48.l.2.5, 96.192.1-48.l.2.6, 96.192.1-48.l.2.7, 96.192.1-48.l.2.8, 240.192.1-48.l.2.1, 240.192.1-48.l.2.2, 240.192.1-48.l.2.3, 240.192.1-48.l.2.4, 240.192.1-48.l.2.5, 240.192.1-48.l.2.6, 240.192.1-48.l.2.7, 240.192.1-48.l.2.8, 240.192.1-48.l.2.9, 240.192.1-48.l.2.10, 240.192.1-48.l.2.11, 240.192.1-48.l.2.12 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x z + z^{2} - w^{2} $ |
| $=$ | $3 x y - 3 y^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + 3 x^{3} y + x^{2} y^{2} + 3 x^{2} z^{2} + 2 x y z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3}\cdot\frac{4251528xz^{21}w^{2}-14880348xz^{19}w^{4}+59718222xz^{15}w^{8}-85056804xz^{13}w^{10}+25771608xz^{11}w^{12}+29060856xz^{9}w^{14}-23119749xz^{7}w^{16}+3724218xz^{5}w^{18}+526122xz^{3}w^{20}+252xzw^{22}-531441z^{24}+4251528z^{22}w^{2}-43381332z^{18}w^{6}+89163990z^{16}w^{8}-35416278z^{14}w^{10}-73099746z^{12}w^{12}+88273152z^{10}w^{14}-27347220z^{8}w^{16}-4925151z^{6}w^{18}+3029049z^{4}w^{20}-20646z^{2}w^{22}-w^{24}}{w^{16}(27xz^{7}-54xz^{5}w^{2}-6xz^{3}w^{4}+28xzw^{6}+27z^{8}+9z^{6}w^{2}-99z^{4}w^{4}+58z^{2}w^{6}+5w^{8})}$ |
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.