Invariants
| Level: | $248$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{6}\cdot4\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}137&204\\40&217\end{bmatrix}$, $\begin{bmatrix}173&40\\204&33\end{bmatrix}$, $\begin{bmatrix}211&48\\116&215\end{bmatrix}$, $\begin{bmatrix}225&78\\40&211\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 248.384.5-248.z.2.1, 248.384.5-248.z.2.2, 248.384.5-248.z.2.3, 248.384.5-248.z.2.4, 248.384.5-248.z.2.5, 248.384.5-248.z.2.6, 248.384.5-248.z.2.7, 248.384.5-248.z.2.8 |
| Cyclic 248-isogeny field degree: | $64$ |
| Cyclic 248-torsion field degree: | $3840$ |
| Full 248-torsion field degree: | $7142400$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.96.1.f.2 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 248.96.1.n.2 | $248$ | $2$ | $2$ | $1$ | $?$ |
| 248.96.1.w.1 | $248$ | $2$ | $2$ | $1$ | $?$ |
| 248.96.3.l.2 | $248$ | $2$ | $2$ | $3$ | $?$ |
| 248.96.3.m.1 | $248$ | $2$ | $2$ | $3$ | $?$ |
| 248.96.3.q.2 | $248$ | $2$ | $2$ | $3$ | $?$ |
| 248.96.3.w.1 | $248$ | $2$ | $2$ | $3$ | $?$ |