$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&16\\36&15\end{bmatrix}$, $\begin{bmatrix}15&38\\9&35\end{bmatrix}$, $\begin{bmatrix}33&38\\29&5\end{bmatrix}$, $\begin{bmatrix}39&30\\19&37\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.gb.2.1, 40.96.1-40.gb.2.2, 40.96.1-40.gb.2.3, 40.96.1-40.gb.2.4, 40.96.1-40.gb.2.5, 40.96.1-40.gb.2.6, 40.96.1-40.gb.2.7, 40.96.1-40.gb.2.8, 80.96.1-40.gb.2.1, 80.96.1-40.gb.2.2, 80.96.1-40.gb.2.3, 80.96.1-40.gb.2.4, 80.96.1-40.gb.2.5, 80.96.1-40.gb.2.6, 80.96.1-40.gb.2.7, 80.96.1-40.gb.2.8, 120.96.1-40.gb.2.1, 120.96.1-40.gb.2.2, 120.96.1-40.gb.2.3, 120.96.1-40.gb.2.4, 120.96.1-40.gb.2.5, 120.96.1-40.gb.2.6, 120.96.1-40.gb.2.7, 120.96.1-40.gb.2.8, 240.96.1-40.gb.2.1, 240.96.1-40.gb.2.2, 240.96.1-40.gb.2.3, 240.96.1-40.gb.2.4, 240.96.1-40.gb.2.5, 240.96.1-40.gb.2.6, 240.96.1-40.gb.2.7, 240.96.1-40.gb.2.8, 280.96.1-40.gb.2.1, 280.96.1-40.gb.2.2, 280.96.1-40.gb.2.3, 280.96.1-40.gb.2.4, 280.96.1-40.gb.2.5, 280.96.1-40.gb.2.6, 280.96.1-40.gb.2.7, 280.96.1-40.gb.2.8 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x y - z^{2} - z w + w^{2} $ |
| $=$ | $10 x^{2} + y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 4 x^{3} z + 5 x^{2} y^{2} - 2 x^{2} z^{2} - 4 x z^{3} + 5 y^{4} + 5 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot5^2\,\frac{9y^{2}z^{10}+288y^{2}z^{9}w-1107y^{2}z^{8}w^{2}+2736y^{2}z^{7}w^{3}-2106y^{2}z^{6}w^{4}-2106y^{2}z^{4}w^{6}-2736y^{2}z^{3}w^{7}-1107y^{2}z^{2}w^{8}-288y^{2}zw^{9}+9y^{2}w^{10}+14z^{12}+132z^{11}w-1026z^{10}w^{2}+3452z^{9}w^{3}-4074z^{8}w^{4}+4200z^{7}w^{5}-1388z^{6}w^{6}-4200z^{5}w^{7}-4074z^{4}w^{8}-3452z^{3}w^{9}-1026z^{2}w^{10}-132zw^{11}+14w^{12}}{(z^{2}+zw-w^{2})^{4}(5y^{2}z^{2}+5y^{2}w^{2}+2z^{4}+4z^{3}w-2z^{2}w^{2}-4zw^{3}+2w^{4})}$ |
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.