Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.0.322 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&3\\36&43\end{bmatrix}$, $\begin{bmatrix}11&41\\40&17\end{bmatrix}$, $\begin{bmatrix}17&14\\12&7\end{bmatrix}$, $\begin{bmatrix}23&22\\40&19\end{bmatrix}$, $\begin{bmatrix}31&45\\28&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.bz.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $24576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 90 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^3}\cdot\frac{(x+y)^{24}(892289439x^{8}+855974304x^{7}y-1149014808x^{6}y^{2}-1239136704x^{5}y^{3}-862173720x^{4}y^{4}-603262080x^{3}y^{5}-140832864x^{2}y^{6}-69408000xy^{7}-10459408y^{8})^{3}}{(x+y)^{26}(9x-2y)^{4}(108x^{2}+18xy+31y^{2})(135x^{2}+72xy+58y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 48.24.0-8.n.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.