| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&39\\20&27\end{bmatrix}$, $\begin{bmatrix}15&13\\32&11\end{bmatrix}$, $\begin{bmatrix}17&37\\0&9\end{bmatrix}$, $\begin{bmatrix}25&22\\4&13\end{bmatrix}$, $\begin{bmatrix}35&6\\8&3\end{bmatrix}$, $\begin{bmatrix}39&22\\32&9\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
40.144.1-40.v.2.1, 40.144.1-40.v.2.2, 40.144.1-40.v.2.3, 40.144.1-40.v.2.4, 40.144.1-40.v.2.5, 40.144.1-40.v.2.6, 40.144.1-40.v.2.7, 40.144.1-40.v.2.8, 40.144.1-40.v.2.9, 40.144.1-40.v.2.10, 40.144.1-40.v.2.11, 40.144.1-40.v.2.12, 40.144.1-40.v.2.13, 40.144.1-40.v.2.14, 40.144.1-40.v.2.15, 40.144.1-40.v.2.16, 80.144.1-40.v.2.1, 80.144.1-40.v.2.2, 80.144.1-40.v.2.3, 80.144.1-40.v.2.4, 80.144.1-40.v.2.5, 80.144.1-40.v.2.6, 80.144.1-40.v.2.7, 80.144.1-40.v.2.8, 80.144.1-40.v.2.9, 80.144.1-40.v.2.10, 80.144.1-40.v.2.11, 80.144.1-40.v.2.12, 80.144.1-40.v.2.13, 80.144.1-40.v.2.14, 80.144.1-40.v.2.15, 80.144.1-40.v.2.16, 80.144.1-40.v.2.17, 80.144.1-40.v.2.18, 80.144.1-40.v.2.19, 80.144.1-40.v.2.20, 80.144.1-40.v.2.21, 80.144.1-40.v.2.22, 80.144.1-40.v.2.23, 80.144.1-40.v.2.24, 120.144.1-40.v.2.1, 120.144.1-40.v.2.2, 120.144.1-40.v.2.3, 120.144.1-40.v.2.4, 120.144.1-40.v.2.5, 120.144.1-40.v.2.6, 120.144.1-40.v.2.7, 120.144.1-40.v.2.8, 120.144.1-40.v.2.9, 120.144.1-40.v.2.10, 120.144.1-40.v.2.11, 120.144.1-40.v.2.12, 120.144.1-40.v.2.13, 120.144.1-40.v.2.14, 120.144.1-40.v.2.15, 120.144.1-40.v.2.16, 240.144.1-40.v.2.1, 240.144.1-40.v.2.2, 240.144.1-40.v.2.3, 240.144.1-40.v.2.4, 240.144.1-40.v.2.5, 240.144.1-40.v.2.6, 240.144.1-40.v.2.7, 240.144.1-40.v.2.8, 240.144.1-40.v.2.9, 240.144.1-40.v.2.10, 240.144.1-40.v.2.11, 240.144.1-40.v.2.12, 240.144.1-40.v.2.13, 240.144.1-40.v.2.14, 240.144.1-40.v.2.15, 240.144.1-40.v.2.16, 240.144.1-40.v.2.17, 240.144.1-40.v.2.18, 240.144.1-40.v.2.19, 240.144.1-40.v.2.20, 240.144.1-40.v.2.21, 240.144.1-40.v.2.22, 240.144.1-40.v.2.23, 240.144.1-40.v.2.24, 280.144.1-40.v.2.1, 280.144.1-40.v.2.2, 280.144.1-40.v.2.3, 280.144.1-40.v.2.4, 280.144.1-40.v.2.5, 280.144.1-40.v.2.6, 280.144.1-40.v.2.7, 280.144.1-40.v.2.8, 280.144.1-40.v.2.9, 280.144.1-40.v.2.10, 280.144.1-40.v.2.11, 280.144.1-40.v.2.12, 280.144.1-40.v.2.13, 280.144.1-40.v.2.14, 280.144.1-40.v.2.15, 280.144.1-40.v.2.16 |
| Cyclic 40-isogeny field degree: |
$2$ |
| Cyclic 40-torsion field degree: |
$32$ |
| Full 40-torsion field degree: |
$10240$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 10 x^{2} - z^{2} - 4 z w $ |
| $=$ | $10 x y + 10 y^{2} + z w - w^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 100 x^{4} - 10 x^{2} y^{2} - 20 x^{2} y z - 20 x^{2} z^{2} + y^{2} z^{2} - 2 y z^{3} + z^{4} $ |
![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 embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{40950y^{2}z^{16}+402300y^{2}z^{15}w-711000y^{2}z^{14}w^{2}-184032000y^{2}z^{13}w^{3}-5152896000y^{2}z^{12}w^{4}-71117769600y^{2}z^{11}w^{5}-579367382400y^{2}z^{10}w^{6}-3062263968000y^{2}z^{9}w^{7}-11143014336000y^{2}z^{8}w^{8}-28905274368000y^{2}z^{7}w^{9}-53918506022400y^{2}z^{6}w^{10}-69979443609600y^{2}z^{5}w^{11}-56701308096000y^{2}z^{4}w^{12}-22684252032000y^{2}z^{3}w^{13}-2845691136000y^{2}z^{2}w^{14}-43381555200y^{2}zw^{15}-7372800y^{2}w^{16}+z^{18}+732z^{17}w+183153z^{16}w^{2}+18750934z^{15}w^{3}+554988060z^{14}w^{4}+8184696432z^{13}w^{5}+72788090064z^{12}w^{6}+431091034176z^{11}w^{7}+1810007491008z^{10}w^{8}+5610474218880z^{9}w^{9}+13137293515008z^{8}w^{10}+23333218354176z^{7}w^{11}+30807649400064z^{6}w^{12}+28541707410432z^{5}w^{13}+16348473250560z^{4}w^{14}+4503987229184z^{3}w^{15}+390028182528z^{2}w^{16}+4522463232zw^{17}+741376w^{18}}{w(z-w)^{5}(z+4w)^{2}(10y^{2}z^{8}-180y^{2}z^{7}w+1480y^{2}z^{6}w^{2}-48160y^{2}z^{5}w^{3}-100800y^{2}z^{4}w^{4}-326560y^{2}z^{3}w^{5}-265120y^{2}z^{2}w^{6}-41280y^{2}zw^{7}-640y^{2}w^{8}-z^{10}+16z^{9}w-113z^{8}w^{2}+438z^{7}w^{3}-836z^{6}w^{4}+3632z^{5}w^{5}+3904z^{4}w^{6}-14192z^{3}w^{7}+4432z^{2}w^{8}+2656zw^{9}+64w^{10})}$ |
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.
