$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}9&35\\8&39\end{bmatrix}$, $\begin{bmatrix}25&30\\28&47\end{bmatrix}$, $\begin{bmatrix}35&6\\40&35\end{bmatrix}$, $\begin{bmatrix}35&20\\12&1\end{bmatrix}$, $\begin{bmatrix}43&17\\24&25\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.bh.1.1, 48.96.1-48.bh.1.2, 48.96.1-48.bh.1.3, 48.96.1-48.bh.1.4, 48.96.1-48.bh.1.5, 48.96.1-48.bh.1.6, 48.96.1-48.bh.1.7, 48.96.1-48.bh.1.8, 48.96.1-48.bh.1.9, 48.96.1-48.bh.1.10, 48.96.1-48.bh.1.11, 48.96.1-48.bh.1.12, 48.96.1-48.bh.1.13, 48.96.1-48.bh.1.14, 48.96.1-48.bh.1.15, 48.96.1-48.bh.1.16, 240.96.1-48.bh.1.1, 240.96.1-48.bh.1.2, 240.96.1-48.bh.1.3, 240.96.1-48.bh.1.4, 240.96.1-48.bh.1.5, 240.96.1-48.bh.1.6, 240.96.1-48.bh.1.7, 240.96.1-48.bh.1.8, 240.96.1-48.bh.1.9, 240.96.1-48.bh.1.10, 240.96.1-48.bh.1.11, 240.96.1-48.bh.1.12, 240.96.1-48.bh.1.13, 240.96.1-48.bh.1.14, 240.96.1-48.bh.1.15, 240.96.1-48.bh.1.16 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$24576$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ - 3 x z + y^{2} + y z + z^{2} $ |
| $=$ | $24 x^{2} + 3 x z + y^{2} + y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 15 x^{2} z^{2} + 11 x z^{3} + 6 y^{2} z^{2} + 7 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^2}{3}\cdot\frac{5970510xz^{11}+7879032xz^{9}w^{2}+6290784xz^{7}w^{4}+1019520xz^{5}w^{6}+59040xz^{3}w^{8}+1152xzw^{10}+1492992z^{12}+995085z^{10}w^{2}-14094z^{8}w^{4}+154224z^{6}w^{6}+26712z^{4}w^{8}+1584z^{2}w^{10}+32w^{12}}{w^{2}z^{2}(162xz^{7}-594xz^{5}w^{2}+576xz^{3}w^{4}-120xzw^{6}+27z^{6}w^{2}-81z^{4}w^{4}+54z^{2}w^{6}-4w^{8})}$ |
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.