$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}11&13\\48&1\end{bmatrix}$, $\begin{bmatrix}17&25\\42&13\end{bmatrix}$, $\begin{bmatrix}17&38\\36&41\end{bmatrix}$, $\begin{bmatrix}17&55\\30&53\end{bmatrix}$, $\begin{bmatrix}41&51\\18&49\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.96.1-60.e.1.1, 60.96.1-60.e.1.2, 60.96.1-60.e.1.3, 60.96.1-60.e.1.4, 60.96.1-60.e.1.5, 60.96.1-60.e.1.6, 60.96.1-60.e.1.7, 60.96.1-60.e.1.8, 60.96.1-60.e.1.9, 60.96.1-60.e.1.10, 60.96.1-60.e.1.11, 60.96.1-60.e.1.12, 120.96.1-60.e.1.1, 120.96.1-60.e.1.2, 120.96.1-60.e.1.3, 120.96.1-60.e.1.4, 120.96.1-60.e.1.5, 120.96.1-60.e.1.6, 120.96.1-60.e.1.7, 120.96.1-60.e.1.8, 120.96.1-60.e.1.9, 120.96.1-60.e.1.10, 120.96.1-60.e.1.11, 120.96.1-60.e.1.12 |
Cyclic 60-isogeny field degree: |
$12$ |
Cyclic 60-torsion field degree: |
$192$ |
Full 60-torsion field degree: |
$46080$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x y - z^{2} $ |
| $=$ | $x^{2} + 9 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 225 x^{4} + x^{2} y^{2} + 50 x^{2} z^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 5w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{(4z^{2}+5w^{2})(93184y^{2}z^{8}-21760y^{2}z^{6}w^{2}-91200y^{2}z^{4}w^{4}-910000y^{2}z^{2}w^{6}-455000y^{2}w^{8}+2048z^{10}-3072z^{8}w^{2}-25920z^{6}w^{4}-196400z^{4}w^{6}-202500z^{2}w^{8}-50625w^{10})}{w^{2}z^{4}(40y^{2}z^{4}-50y^{2}z^{2}w^{2}-125y^{2}w^{4}+8z^{6}-5z^{4}w^{2})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.