$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&10\\8&13\end{bmatrix}$, $\begin{bmatrix}15&13\\20&17\end{bmatrix}$, $\begin{bmatrix}19&5\\14&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-24.hs.1.1, 48.96.1-24.hs.1.2, 48.96.1-24.hs.1.3, 48.96.1-24.hs.1.4, 48.96.1-24.hs.1.5, 48.96.1-24.hs.1.6, 240.96.1-24.hs.1.1, 240.96.1-24.hs.1.2, 240.96.1-24.hs.1.3, 240.96.1-24.hs.1.4, 240.96.1-24.hs.1.5, 240.96.1-24.hs.1.6 |
Cyclic 24-isogeny field degree: |
$16$ |
Cyclic 24-torsion field degree: |
$128$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 y^{2} + 2 y z + 5 z^{2} - 2 w^{2} $ |
| $=$ | $12 x^{2} + y^{2} + y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 6 x^{2} z^{2} + 25 y^{4} + 60 y^{2} z^{2} + 36 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\cdot3^3\,\frac{2173878yz^{11}+1360800yz^{9}w^{2}-6671700yz^{7}w^{4}+2247000yz^{5}w^{6}-106250yz^{3}w^{8}+75000yzw^{10}+1242945z^{12}-6353478z^{10}w^{2}+752895z^{8}w^{4}+2592100z^{6}w^{6}-1298125z^{4}w^{8}+206250z^{2}w^{10}+3125w^{12}}{8695512yz^{11}-19415700yz^{9}w^{2}+2567700yz^{7}w^{4}+4698000yz^{5}w^{6}-1800000yz^{3}w^{8}+300000yzw^{10}+4971780z^{12}+8120088z^{10}w^{2}-23210145z^{8}w^{4}+12897900z^{6}w^{6}-3217500z^{4}w^{8}+450000z^{2}w^{10}-50000w^{12}}$ |
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.