$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}1&12\\14&39\end{bmatrix}$, $\begin{bmatrix}7&20\\0&1\end{bmatrix}$, $\begin{bmatrix}13&8\\6&25\end{bmatrix}$, $\begin{bmatrix}25&32\\34&17\end{bmatrix}$, $\begin{bmatrix}27&32\\16&13\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.bc.2.1, 40.96.1-40.bc.2.2, 40.96.1-40.bc.2.3, 40.96.1-40.bc.2.4, 40.96.1-40.bc.2.5, 40.96.1-40.bc.2.6, 40.96.1-40.bc.2.7, 40.96.1-40.bc.2.8, 40.96.1-40.bc.2.9, 40.96.1-40.bc.2.10, 40.96.1-40.bc.2.11, 40.96.1-40.bc.2.12, 40.96.1-40.bc.2.13, 40.96.1-40.bc.2.14, 40.96.1-40.bc.2.15, 40.96.1-40.bc.2.16, 120.96.1-40.bc.2.1, 120.96.1-40.bc.2.2, 120.96.1-40.bc.2.3, 120.96.1-40.bc.2.4, 120.96.1-40.bc.2.5, 120.96.1-40.bc.2.6, 120.96.1-40.bc.2.7, 120.96.1-40.bc.2.8, 120.96.1-40.bc.2.9, 120.96.1-40.bc.2.10, 120.96.1-40.bc.2.11, 120.96.1-40.bc.2.12, 120.96.1-40.bc.2.13, 120.96.1-40.bc.2.14, 120.96.1-40.bc.2.15, 120.96.1-40.bc.2.16, 280.96.1-40.bc.2.1, 280.96.1-40.bc.2.2, 280.96.1-40.bc.2.3, 280.96.1-40.bc.2.4, 280.96.1-40.bc.2.5, 280.96.1-40.bc.2.6, 280.96.1-40.bc.2.7, 280.96.1-40.bc.2.8, 280.96.1-40.bc.2.9, 280.96.1-40.bc.2.10, 280.96.1-40.bc.2.11, 280.96.1-40.bc.2.12, 280.96.1-40.bc.2.13, 280.96.1-40.bc.2.14, 280.96.1-40.bc.2.15, 280.96.1-40.bc.2.16 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x z + y^{2} + z^{2} $ |
| $=$ | $5 x^{2} - 3 x z + 2 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{2} z^{2} + 5 y^{2} z^{2} + 2 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^8}{5^2}\cdot\frac{18750xz^{9}w^{2}-7500xz^{7}w^{4}+2500xz^{5}w^{6}-500xz^{3}w^{8}+30xzw^{10}-15625z^{12}-1875z^{8}w^{4}+1500z^{6}w^{6}-275z^{4}w^{8}+60z^{2}w^{10}-w^{12}}{w^{4}z^{4}(100xz^{3}-20xzw^{2}-20z^{2}w^{2}+w^{4})}$ |
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.