$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&20\\10&21\end{bmatrix}$, $\begin{bmatrix}11&10\\10&21\end{bmatrix}$, $\begin{bmatrix}15&4\\8&15\end{bmatrix}$, $\begin{bmatrix}23&2\\2&21\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.bq.1.1, 24.96.1-24.bq.1.2, 24.96.1-24.bq.1.3, 24.96.1-24.bq.1.4, 120.96.1-24.bq.1.1, 120.96.1-24.bq.1.2, 120.96.1-24.bq.1.3, 120.96.1-24.bq.1.4, 168.96.1-24.bq.1.1, 168.96.1-24.bq.1.2, 168.96.1-24.bq.1.3, 168.96.1-24.bq.1.4, 264.96.1-24.bq.1.1, 264.96.1-24.bq.1.2, 264.96.1-24.bq.1.3, 264.96.1-24.bq.1.4, 312.96.1-24.bq.1.1, 312.96.1-24.bq.1.2, 312.96.1-24.bq.1.3, 312.96.1-24.bq.1.4 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y - x z + y^{2} + z^{2} $ |
| $=$ | $x^{2} + x y + 2 x z - x w + y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + x^{3} y - 3 x^{3} z + x^{2} y^{2} - x^{2} y z + 5 x^{2} z^{2} - 2 x y^{2} z + x y z^{2} + \cdots + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{63xz^{11}+321xz^{10}w+891xz^{9}w^{2}+1089xz^{8}w^{3}+318xz^{7}w^{4}-558xz^{6}w^{5}-714xz^{5}w^{6}-390xz^{4}w^{7}-108xz^{3}w^{8}-12xz^{2}w^{9}+64z^{12}+21z^{10}w^{2}+114z^{9}w^{3}+252z^{8}w^{4}+432z^{7}w^{5}+278z^{6}w^{6}-36z^{5}w^{7}-159z^{4}w^{8}-124z^{3}w^{9}-54z^{2}w^{10}-12zw^{11}-w^{12}}{z^{8}(3xz^{3}+5xz^{2}w+3xzw^{2}+xw^{3}+z^{2}w^{2}+2zw^{3})}$ |
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.