| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}45&4\\47&15\end{bmatrix}$, $\begin{bmatrix}45&44\\52&5\end{bmatrix}$, $\begin{bmatrix}51&5\\32&29\end{bmatrix}$, $\begin{bmatrix}51&29\\4&29\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.320.9-60.s.1.1, 60.320.9-60.s.1.2, 60.320.9-60.s.1.3, 60.320.9-60.s.1.4, 120.320.9-60.s.1.1, 120.320.9-60.s.1.2, 120.320.9-60.s.1.3, 120.320.9-60.s.1.4 |
| Cyclic 60-isogeny field degree: |
$72$ |
| Cyclic 60-torsion field degree: |
$1152$ |
| Full 60-torsion field degree: |
$13824$ |
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ 2 x^{2} + x y - x z - x w + 2 x t - x s - y^{2} + y z - y w - y t - y s + z w + z u - z r - t v + \cdots + v r $ |
| $=$ | $x y - x t + x u + x s + y^{2} + 2 y t + y u + y s - z w - z u - z v - z r + 2 z s + t r + t s + u v$ |
| $=$ | $x^{2} + 2 x y + 2 x t + x u + y^{2} + 2 y t + y u + t u + t v + u^{2} - v^{2}$ |
| $=$ | $2 x^{2} - x y + x z - x w + x t + 2 x u + x v + x r + x s + 2 y z + y t - y u - y v - z w + z t + \cdots + s^{2}$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 10000 x^{8} y^{4} - 300000 x^{8} y^{2} z^{2} + 450000 x^{8} z^{4} + 12000 x^{7} y^{5} + \cdots + 441045 z^{12} $ |
![Copy content]()
![Toggle raw display]()
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 canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle s$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}r$ |
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
20.80.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y-w+t$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle -z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ XZ-YZ+YW-2ZW+W^{2}-3XT-3YT+2ZT-WT $ |
|
$=$ |
$ X^{2}+3XY+Y^{2}-5XZ-5YZ+5Z^{2}-XW+6YW-5ZW-W^{2}+3XT-3YT+WT+T^{2} $ |
|
$=$ |
$ 9XY-XZ-YZ-Z^{2}+4XW-3YW+ZW+2W^{2}-3XT+3YT-WT-3T^{2} $ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
