$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}25&14\\21&25\end{bmatrix}$, $\begin{bmatrix}27&40\\46&23\end{bmatrix}$, $\begin{bmatrix}33&53\\28&47\end{bmatrix}$, $\begin{bmatrix}41&1\\42&49\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.80.1-60.f.1.1, 60.80.1-60.f.1.2, 60.80.1-60.f.1.3, 60.80.1-60.f.1.4, 120.80.1-60.f.1.1, 120.80.1-60.f.1.2, 120.80.1-60.f.1.3, 120.80.1-60.f.1.4 |
Cyclic 60-isogeny field degree: |
$144$ |
Cyclic 60-torsion field degree: |
$2304$ |
Full 60-torsion field degree: |
$55296$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x^{2} - 3 y^{2} - z^{2} - w^{2} $ |
| $=$ | $3 x^{2} + 6 y^{2} + z^{2} - z w + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} + 15 x^{2} z^{2} + 9 y^{4} + 30 y^{2} z^{2} + 45 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 5^3\,\frac{(2z-w)(3z-w)^{3}(2z^{2}-3zw+3w^{2})^{3}}{(z^{2}+zw-w^{2})^{5}}$ |
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.