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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}35&60\\222&7\end{bmatrix}$, $\begin{bmatrix}139&20\\88&85\end{bmatrix}$, $\begin{bmatrix}141&76\\72&231\end{bmatrix}$, $\begin{bmatrix}191&40\\2&103\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 248.48.0.r.1 for the level structure with $-I$) |
| Cyclic 248-isogeny field degree: | $64$ |
| Cyclic 248-torsion field degree: | $7680$ |
| 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.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 248.48.0-8.e.1.16 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.i.1.18 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.i.1.31 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.l.1.6 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.l.1.14 | $248$ | $2$ | $2$ | $0$ | $?$ |
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.r.1.7 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.u.2.1 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.w.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.z.2.5 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.be.1.5 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bf.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bg.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bh.1.3 | $248$ | $2$ | $2$ | $1$ |