| $\GL_2(\Z/32\Z)$-generators: |
$\begin{bmatrix}3&18\\16&15\end{bmatrix}$, $\begin{bmatrix}25&29\\0&21\end{bmatrix}$, $\begin{bmatrix}27&16\\0&5\end{bmatrix}$, $\begin{bmatrix}29&29\\16&5\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
32.192.1-32.e.2.1, 32.192.1-32.e.2.2, 32.192.1-32.e.2.3, 32.192.1-32.e.2.4, 32.192.1-32.e.2.5, 32.192.1-32.e.2.6, 32.192.1-32.e.2.7, 32.192.1-32.e.2.8, 96.192.1-32.e.2.1, 96.192.1-32.e.2.2, 96.192.1-32.e.2.3, 96.192.1-32.e.2.4, 96.192.1-32.e.2.5, 96.192.1-32.e.2.6, 96.192.1-32.e.2.7, 96.192.1-32.e.2.8, 160.192.1-32.e.2.1, 160.192.1-32.e.2.2, 160.192.1-32.e.2.3, 160.192.1-32.e.2.4, 160.192.1-32.e.2.5, 160.192.1-32.e.2.6, 160.192.1-32.e.2.7, 160.192.1-32.e.2.8, 224.192.1-32.e.2.1, 224.192.1-32.e.2.2, 224.192.1-32.e.2.3, 224.192.1-32.e.2.4, 224.192.1-32.e.2.5, 224.192.1-32.e.2.6, 224.192.1-32.e.2.7, 224.192.1-32.e.2.8 |
| Cyclic 32-isogeny field degree: |
$2$ |
| Cyclic 32-torsion field degree: |
$16$ |
| Full 32-torsion field degree: |
$4096$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y - z^{2} $ |
| $=$ | $5 x^{2} + 5 x y - 2 y^{2} + z^{2} - w^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 4 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{5}{2^4}\cdot\frac{37748736y^{2}z^{20}w^{2}+869400576y^{2}z^{16}w^{6}+255393792y^{2}z^{12}w^{10}+13842432y^{2}z^{8}w^{14}+262080y^{2}z^{4}w^{18}+1638y^{2}w^{22}-4194304z^{24}+75497472z^{22}w^{2}-727449600z^{20}w^{4}+1738801152z^{18}w^{6}-1453572096z^{16}w^{8}+510787584z^{14}w^{10}-136523776z^{12}w^{12}+27684864z^{10}w^{14}-4089408z^{8}w^{16}+524160z^{6}w^{18}-49188z^{4}w^{20}+3276z^{2}w^{22}-205w^{24}}{w^{2}z^{8}(16384y^{2}z^{12}+4608y^{2}z^{8}w^{4}+192y^{2}z^{4}w^{8}+2y^{2}w^{12}+32768z^{14}+33792z^{12}w^{2}+9216z^{10}w^{4}+3904z^{8}w^{6}+384z^{6}w^{8}+116z^{4}w^{10}+4z^{2}w^{12}+w^{14})}$ |
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.
