$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&34\\26&33\end{bmatrix}$, $\begin{bmatrix}12&9\\33&3\end{bmatrix}$, $\begin{bmatrix}17&9\\38&13\end{bmatrix}$, $\begin{bmatrix}36&5\\3&38\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.48.1-40.cj.1.1, 40.48.1-40.cj.1.2, 40.48.1-40.cj.1.3, 40.48.1-40.cj.1.4, 40.48.1-40.cj.1.5, 40.48.1-40.cj.1.6, 40.48.1-40.cj.1.7, 40.48.1-40.cj.1.8, 120.48.1-40.cj.1.1, 120.48.1-40.cj.1.2, 120.48.1-40.cj.1.3, 120.48.1-40.cj.1.4, 120.48.1-40.cj.1.5, 120.48.1-40.cj.1.6, 120.48.1-40.cj.1.7, 120.48.1-40.cj.1.8, 280.48.1-40.cj.1.1, 280.48.1-40.cj.1.2, 280.48.1-40.cj.1.3, 280.48.1-40.cj.1.4, 280.48.1-40.cj.1.5, 280.48.1-40.cj.1.6, 280.48.1-40.cj.1.7, 280.48.1-40.cj.1.8 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$96$ |
Full 40-torsion field degree: |
$30720$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - x^{2} - 4133x + 103637 $ |
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 24 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^5\cdot5^5}\cdot\frac{80x^{2}y^{6}-15600000x^{2}y^{4}z^{2}+942640000000x^{2}y^{2}z^{4}-18756240000000000x^{2}z^{6}-7520xy^{6}z+1245600000xy^{4}z^{3}-72102560000000xy^{2}z^{5}+1404875360000000000xz^{7}-y^{8}+308720y^{6}z^{2}-33152800000y^{4}z^{4}+1546420560000000y^{2}z^{6}-26303096760000000000z^{8}}{z^{3}y^{2}(12200x^{2}z+xy^{2}-913800xz^{2}-147y^{2}z+17108800z^{3})}$ |
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.