$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&32\\24&7\end{bmatrix}$, $\begin{bmatrix}7&4\\16&31\end{bmatrix}$, $\begin{bmatrix}9&26\\4&15\end{bmatrix}$, $\begin{bmatrix}17&20\\24&1\end{bmatrix}$, $\begin{bmatrix}27&10\\32&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.3-40.l.1.1, 40.192.3-40.l.1.2, 40.192.3-40.l.1.3, 40.192.3-40.l.1.4, 40.192.3-40.l.1.5, 40.192.3-40.l.1.6, 40.192.3-40.l.1.7, 40.192.3-40.l.1.8, 40.192.3-40.l.1.9, 40.192.3-40.l.1.10, 40.192.3-40.l.1.11, 40.192.3-40.l.1.12, 120.192.3-40.l.1.1, 120.192.3-40.l.1.2, 120.192.3-40.l.1.3, 120.192.3-40.l.1.4, 120.192.3-40.l.1.5, 120.192.3-40.l.1.6, 120.192.3-40.l.1.7, 120.192.3-40.l.1.8, 120.192.3-40.l.1.9, 120.192.3-40.l.1.10, 120.192.3-40.l.1.11, 120.192.3-40.l.1.12, 280.192.3-40.l.1.1, 280.192.3-40.l.1.2, 280.192.3-40.l.1.3, 280.192.3-40.l.1.4, 280.192.3-40.l.1.5, 280.192.3-40.l.1.6, 280.192.3-40.l.1.7, 280.192.3-40.l.1.8, 280.192.3-40.l.1.9, 280.192.3-40.l.1.10, 280.192.3-40.l.1.11, 280.192.3-40.l.1.12 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ y w + z t $ |
| $=$ | $2 x w + y w + z u$ |
| $=$ | $ - x t - x u - y t + 2 z w$ |
| $=$ | $2 x t + y t - y u$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 100 x^{6} z^{2} + x^{4} y^{4} - 40 x^{4} y^{2} z^{2} + 200 x^{4} z^{4} + 3 x^{2} y^{4} z^{2} + \cdots + 2 y^{4} z^{4} $ |
Geometric Weierstrass model Geometric Weierstrass model
$ w^{2} $ | $=$ | $ -10 x^{2} y z - 20 y z^{3} $ |
$0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 5z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}u$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{300000y^{2}z^{8}u^{2}+165000y^{2}z^{6}u^{4}+67000y^{2}z^{4}u^{6}+66050y^{2}z^{2}u^{8}+47910y^{2}u^{10}-1000000z^{12}-600000z^{10}u^{2}-30000z^{8}u^{4}+10000z^{6}u^{6}-1500z^{4}u^{8}+2260z^{2}u^{10}-t^{12}-6t^{11}u-21t^{10}u^{2}-56t^{9}u^{3}-168t^{8}u^{4}-504t^{7}u^{5}-1343t^{6}u^{6}-3114t^{5}u^{7}-6843t^{4}u^{8}-13980t^{3}u^{9}-22996t^{2}u^{10}-11832tu^{11}+1816u^{12}}{u^{4}(750y^{2}z^{2}u^{4}+670y^{2}u^{6}-100z^{4}u^{4}+20z^{2}u^{6}-t^{8}-6t^{7}u-21t^{6}u^{2}-56t^{5}u^{3}-122t^{4}u^{4}-228t^{3}u^{5}-347t^{2}u^{6}-178tu^{7}+23u^{8})}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.