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: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}53&232\\176&109\end{bmatrix}$, $\begin{bmatrix}81&12\\152&23\end{bmatrix}$, $\begin{bmatrix}89&40\\238&153\end{bmatrix}$, $\begin{bmatrix}125&156\\110&231\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 248.48.0.ba.2 for the level structure with $-I$) |
| Cyclic 248-isogeny field degree: | $32$ |
| 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 infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 248.48.0-248.h.2.5 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.h.2.13 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-8.i.1.5 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.i.1.1 | $248$ | $2$ | $2$ | $0$ | $?$ |
| 248.48.0-248.i.1.5 | $248$ | $2$ | $2$ | $0$ | $?$ |
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.f.2.2 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.q.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.bq.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.br.2.3 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.ca.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.cd.2.3 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.ce.2.2 | $248$ | $2$ | $2$ | $1$ |
| 248.192.1-248.ch.1.1 | $248$ | $2$ | $2$ | $1$ |