$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}5&4\\16&9\end{bmatrix}$, $\begin{bmatrix}23&21\\4&11\end{bmatrix}$, $\begin{bmatrix}31&13\\12&17\end{bmatrix}$, $\begin{bmatrix}39&40\\40&15\end{bmatrix}$, $\begin{bmatrix}41&30\\16&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.48.1-48.d.1.1, 48.48.1-48.d.1.2, 48.48.1-48.d.1.3, 48.48.1-48.d.1.4, 48.48.1-48.d.1.5, 48.48.1-48.d.1.6, 48.48.1-48.d.1.7, 48.48.1-48.d.1.8, 48.48.1-48.d.1.9, 48.48.1-48.d.1.10, 48.48.1-48.d.1.11, 48.48.1-48.d.1.12, 48.48.1-48.d.1.13, 48.48.1-48.d.1.14, 48.48.1-48.d.1.15, 48.48.1-48.d.1.16, 240.48.1-48.d.1.1, 240.48.1-48.d.1.2, 240.48.1-48.d.1.3, 240.48.1-48.d.1.4, 240.48.1-48.d.1.5, 240.48.1-48.d.1.6, 240.48.1-48.d.1.7, 240.48.1-48.d.1.8, 240.48.1-48.d.1.9, 240.48.1-48.d.1.10, 240.48.1-48.d.1.11, 240.48.1-48.d.1.12, 240.48.1-48.d.1.13, 240.48.1-48.d.1.14, 240.48.1-48.d.1.15, 240.48.1-48.d.1.16 |
Cyclic 48-isogeny field degree: |
$16$ |
Cyclic 48-torsion field degree: |
$256$ |
Full 48-torsion field degree: |
$49152$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + 36x $ |
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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3^8}\cdot\frac{243x^{2}y^{4}z^{2}-36xy^{6}z+19683xy^{2}z^{5}+y^{8}+531441z^{8}}{z^{5}y^{2}x}$ |
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.