$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&7\\20&15\end{bmatrix}$, $\begin{bmatrix}1&22\\16&21\end{bmatrix}$, $\begin{bmatrix}1&23\\8&23\end{bmatrix}$, $\begin{bmatrix}15&10\\4&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.fa.1.1, 24.96.1-24.fa.1.2, 24.96.1-24.fa.1.3, 24.96.1-24.fa.1.4, 48.96.1-24.fa.1.1, 48.96.1-24.fa.1.2, 48.96.1-24.fa.1.3, 48.96.1-24.fa.1.4, 120.96.1-24.fa.1.1, 120.96.1-24.fa.1.2, 120.96.1-24.fa.1.3, 120.96.1-24.fa.1.4, 168.96.1-24.fa.1.1, 168.96.1-24.fa.1.2, 168.96.1-24.fa.1.3, 168.96.1-24.fa.1.4, 240.96.1-24.fa.1.1, 240.96.1-24.fa.1.2, 240.96.1-24.fa.1.3, 240.96.1-24.fa.1.4, 264.96.1-24.fa.1.1, 264.96.1-24.fa.1.2, 264.96.1-24.fa.1.3, 264.96.1-24.fa.1.4, 312.96.1-24.fa.1.1, 312.96.1-24.fa.1.2, 312.96.1-24.fa.1.3, 312.96.1-24.fa.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^{2} - x z + 6 y^{2} + z^{2} $ |
| $=$ | $7 x^{2} - 4 x z + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 20 x^{2} y^{2} + 3 x^{2} z^{2} + 196 y^{4} + 84 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^2}\cdot\frac{30195296640xz^{11}-63407183616xz^{9}w^{2}+28300176576xz^{7}w^{4}+385850304xz^{5}w^{6}-642565224xz^{3}w^{8}+121010400xzw^{10}-9922751424z^{12}+51334523712z^{10}w^{2}-59671379376z^{8}w^{4}+25404182496z^{6}w^{6}-4834279044z^{4}w^{8}+230351940z^{2}w^{10}+2100875w^{12}}{31065120xz^{11}+7764120xz^{9}w^{2}-12674340xz^{7}w^{4}+1382976xz^{5}w^{6}+941192xz^{3}w^{8}-67228xzw^{10}-10208592z^{12}-24629832z^{10}w^{2}+5387823z^{8}w^{4}+2916186z^{6}w^{6}-929187z^{4}w^{8}+4802z^{2}w^{10}+16807w^{12}}$ |
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.