$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}11&39\\8&25\end{bmatrix}$, $\begin{bmatrix}13&15\\0&31\end{bmatrix}$, $\begin{bmatrix}15&22\\16&43\end{bmatrix}$, $\begin{bmatrix}23&21\\8&1\end{bmatrix}$, $\begin{bmatrix}29&35\\20&25\end{bmatrix}$, $\begin{bmatrix}35&25\\4&43\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.288.8-48.iv.2.1, 48.288.8-48.iv.2.2, 48.288.8-48.iv.2.3, 48.288.8-48.iv.2.4, 48.288.8-48.iv.2.5, 48.288.8-48.iv.2.6, 48.288.8-48.iv.2.7, 48.288.8-48.iv.2.8, 48.288.8-48.iv.2.9, 48.288.8-48.iv.2.10, 48.288.8-48.iv.2.11, 48.288.8-48.iv.2.12, 48.288.8-48.iv.2.13, 48.288.8-48.iv.2.14, 48.288.8-48.iv.2.15, 48.288.8-48.iv.2.16, 48.288.8-48.iv.2.17, 48.288.8-48.iv.2.18, 48.288.8-48.iv.2.19, 48.288.8-48.iv.2.20, 48.288.8-48.iv.2.21, 48.288.8-48.iv.2.22, 48.288.8-48.iv.2.23, 48.288.8-48.iv.2.24, 48.288.8-48.iv.2.25, 48.288.8-48.iv.2.26, 48.288.8-48.iv.2.27, 48.288.8-48.iv.2.28, 48.288.8-48.iv.2.29, 48.288.8-48.iv.2.30, 48.288.8-48.iv.2.31, 48.288.8-48.iv.2.32, 96.288.8-48.iv.2.1, 96.288.8-48.iv.2.2, 96.288.8-48.iv.2.3, 96.288.8-48.iv.2.4, 96.288.8-48.iv.2.5, 96.288.8-48.iv.2.6, 96.288.8-48.iv.2.7, 96.288.8-48.iv.2.8, 96.288.8-48.iv.2.9, 96.288.8-48.iv.2.10, 96.288.8-48.iv.2.11, 96.288.8-48.iv.2.12, 96.288.8-48.iv.2.13, 96.288.8-48.iv.2.14, 96.288.8-48.iv.2.15, 96.288.8-48.iv.2.16, 240.288.8-48.iv.2.1, 240.288.8-48.iv.2.2, 240.288.8-48.iv.2.3, 240.288.8-48.iv.2.4, 240.288.8-48.iv.2.5, 240.288.8-48.iv.2.6, 240.288.8-48.iv.2.7, 240.288.8-48.iv.2.8, 240.288.8-48.iv.2.9, 240.288.8-48.iv.2.10, 240.288.8-48.iv.2.11, 240.288.8-48.iv.2.12, 240.288.8-48.iv.2.13, 240.288.8-48.iv.2.14, 240.288.8-48.iv.2.15, 240.288.8-48.iv.2.16, 240.288.8-48.iv.2.17, 240.288.8-48.iv.2.18, 240.288.8-48.iv.2.19, 240.288.8-48.iv.2.20, 240.288.8-48.iv.2.21, 240.288.8-48.iv.2.22, 240.288.8-48.iv.2.23, 240.288.8-48.iv.2.24, 240.288.8-48.iv.2.25, 240.288.8-48.iv.2.26, 240.288.8-48.iv.2.27, 240.288.8-48.iv.2.28, 240.288.8-48.iv.2.29, 240.288.8-48.iv.2.30, 240.288.8-48.iv.2.31, 240.288.8-48.iv.2.32 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$8192$ |
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
$ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $u^{2} - u v + v^{2} + r^{2}$ |
| $=$ | $x r + y u - y v + t v$ |
| $=$ | $x r - y u + t u - t v$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{12} + 33 x^{10} z^{2} + 387 x^{8} z^{4} + 27 x^{6} y^{3} z^{3} + 1971 x^{6} z^{6} + \cdots + 1728 y^{6} z^{6} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle u$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}r$ |
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
24.72.4.gk.2
:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle -u$ |
$\displaystyle W$ |
$=$ |
$\displaystyle -v$ |
Equation of the image curve:
$0$ |
$=$ |
$ XZ-YZ+XW+2YW $ |
|
$=$ |
$ 7X^{3}+4X^{2}Y+4XY^{2}-Z^{3}+Z^{2}W-ZW^{2} $ |
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.