Invariants
| Level: | $48$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.24.0.26 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&31\\14&21\end{bmatrix}$, $\begin{bmatrix}3&8\\20&29\end{bmatrix}$, $\begin{bmatrix}17&9\\30&11\end{bmatrix}$, $\begin{bmatrix}27&32\\8&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.i.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| Full 48-torsion field degree: | $49152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 33 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(x-4y)^{12}(5x^{4}-32x^{3}y+64x^{2}y^{2}-1024xy^{3}+5120y^{4})^{3}}{(x-4y)^{12}(x^{2}-32y^{2})^{2}(x^{2}-16xy+32y^{2})^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 48.48.1-8.k.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-8.k.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.k.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.k.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-8.l.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-8.l.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.l.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.l.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.72.2-24.ba.1.13 | $48$ | $3$ | $3$ | $2$ |
| 48.96.1-24.dr.1.5 | $48$ | $4$ | $4$ | $1$ |
| 240.48.1-40.k.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-40.k.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.k.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.k.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-40.l.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-40.l.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.l.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.l.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.120.4-40.o.1.6 | $240$ | $5$ | $5$ | $4$ |
| 240.144.3-40.u.1.14 | $240$ | $6$ | $6$ | $3$ |
| 240.240.7-40.ba.1.9 | $240$ | $10$ | $10$ | $7$ |