$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}13&5\\2&29\end{bmatrix}$, $\begin{bmatrix}16&55\\9&49\end{bmatrix}$, $\begin{bmatrix}17&50\\11&13\end{bmatrix}$, $\begin{bmatrix}32&25\\15&49\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.48.1-60.f.1.1, 60.48.1-60.f.1.2, 60.48.1-60.f.1.3, 60.48.1-60.f.1.4, 120.48.1-60.f.1.1, 120.48.1-60.f.1.2, 120.48.1-60.f.1.3, 120.48.1-60.f.1.4 |
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 $ | $=$ | $ 15 x^{2} - y w $ |
| $=$ | $125 y^{2} - 22 y w - 15 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} - 66 x^{2} z^{2} - 3 y^{2} z^{2} + 45 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{15}w$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{89062500yz^{4}w+869820000yz^{2}w^{3}+4376384yw^{5}-1953125z^{6}-171825000z^{4}w^{2}-119227920z^{2}w^{4}-329472w^{6}}{w(3515625yz^{4}+2253750yz^{2}w^{2}+68381yw^{4}+1237500z^{4}w-5280z^{2}w^{3}-5148w^{5})}$ |
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.