$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&17\\12&19\end{bmatrix}$, $\begin{bmatrix}7&5\\6&5\end{bmatrix}$, $\begin{bmatrix}23&5\\0&17\end{bmatrix}$, $\begin{bmatrix}23&21\\12&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.ci.1.1, 24.96.1-24.ci.1.2, 24.96.1-24.ci.1.3, 24.96.1-24.ci.1.4, 24.96.1-24.ci.1.5, 24.96.1-24.ci.1.6, 24.96.1-24.ci.1.7, 24.96.1-24.ci.1.8, 120.96.1-24.ci.1.1, 120.96.1-24.ci.1.2, 120.96.1-24.ci.1.3, 120.96.1-24.ci.1.4, 120.96.1-24.ci.1.5, 120.96.1-24.ci.1.6, 120.96.1-24.ci.1.7, 120.96.1-24.ci.1.8, 168.96.1-24.ci.1.1, 168.96.1-24.ci.1.2, 168.96.1-24.ci.1.3, 168.96.1-24.ci.1.4, 168.96.1-24.ci.1.5, 168.96.1-24.ci.1.6, 168.96.1-24.ci.1.7, 168.96.1-24.ci.1.8, 264.96.1-24.ci.1.1, 264.96.1-24.ci.1.2, 264.96.1-24.ci.1.3, 264.96.1-24.ci.1.4, 264.96.1-24.ci.1.5, 264.96.1-24.ci.1.6, 264.96.1-24.ci.1.7, 264.96.1-24.ci.1.8, 312.96.1-24.ci.1.1, 312.96.1-24.ci.1.2, 312.96.1-24.ci.1.3, 312.96.1-24.ci.1.4, 312.96.1-24.ci.1.5, 312.96.1-24.ci.1.6, 312.96.1-24.ci.1.7, 312.96.1-24.ci.1.8 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x y - z^{2} $ |
| $=$ | $2 x^{2} + 2 x y + 18 y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} + 6 x^{2} y^{2} + 10 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^3\cdot3}\cdot\frac{(8z^{2}+3w^{2})(8945664y^{2}z^{8}-626688y^{2}z^{6}w^{2}-787968y^{2}z^{4}w^{4}-2358720y^{2}z^{2}w^{6}-353808y^{2}w^{8}+327680z^{10}-147456z^{8}w^{2}-373248z^{6}w^{4}-848448z^{4}w^{6}-262440z^{2}w^{8}-19683w^{10})}{w^{2}z^{4}(96y^{2}z^{4}-36y^{2}z^{2}w^{2}-27y^{2}w^{4}+32z^{6}-6z^{4}w^{2})}$ |
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.