Invariants
| Level: | $248$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}25&116\\112&85\end{bmatrix}$, $\begin{bmatrix}169&220\\4&197\end{bmatrix}$, $\begin{bmatrix}185&92\\244&167\end{bmatrix}$, $\begin{bmatrix}247&214\\116&83\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 248.48.0.i.1 for the level structure with $-I$) |
| Cyclic 248-isogeny field degree: | $64$ |
| Cyclic 248-torsion field degree: | $3840$ |
| Full 248-torsion field degree: | $14284800$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 124.48.0-124.c.1.3 | $124$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-124.c.1.14 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-8.e.2.1 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.h.2.2 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.h.2.28 | $248$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 248.192.1-248.i.2.5 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.y.2.5 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bc.2.6 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bg.2.7 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bu.2.6 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.by.2.7 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.cb.2.7 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.cd.2.6 | $248$ | $2$ | $2$ | $1$ |