$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}14&55\\27&43\end{bmatrix}$, $\begin{bmatrix}22&25\\25&59\end{bmatrix}$, $\begin{bmatrix}58&5\\33&58\end{bmatrix}$, $\begin{bmatrix}58&45\\37&52\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.48.1-60.f.2.1, 60.48.1-60.f.2.2, 60.48.1-60.f.2.3, 60.48.1-60.f.2.4, 120.48.1-60.f.2.1, 120.48.1-60.f.2.2, 120.48.1-60.f.2.3, 120.48.1-60.f.2.4 |
Cyclic 60-isogeny field degree: |
$24$ |
Cyclic 60-torsion field degree: |
$384$ |
Full 60-torsion field degree: |
$92160$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x^{2} - 5 y^{2} + 4 y w + 3 z^{2} - w^{2} $ |
| $=$ | $9 x^{2} + 5 y^{2} - 5 y w - 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 125 x^{4} - 66 x^{2} z^{2} - 3 y^{2} z^{2} + 9 z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{3562500yz^{4}w+173964000yz^{2}w^{3}+4376384yw^{5}-78125z^{6}-34365000z^{4}w^{2}-119227920z^{2}w^{4}-1647360w^{6}}{w(140625yz^{4}+450750yz^{2}w^{2}+68381yw^{4}+247500z^{4}w-5280z^{2}w^{3}-25740w^{5})}$ |
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.