Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 16H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.451 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&42\\0&29\end{bmatrix}$, $\begin{bmatrix}17&36\\28&31\end{bmatrix}$, $\begin{bmatrix}19&35\\20&23\end{bmatrix}$, $\begin{bmatrix}41&32\\8&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bs.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $12288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3}\cdot\frac{x^{48}(429981696x^{16}+4299816960x^{14}y^{2}+1612431360x^{12}y^{4}+209018880x^{10}y^{6}+22685184x^{8}y^{8}+1451520x^{6}y^{10}+77760x^{4}y^{12}+1440x^{2}y^{14}+y^{16})^{3}}{y^{2}x^{50}(12x^{2}-y^{2})^{16}(12x^{2}+y^{2})^{4}(144x^{4}+y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.g.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.e.1.17 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.g.1.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.