Invariants
| Level: | $248$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}5&68\\237&123\end{bmatrix}$, $\begin{bmatrix}57&120\\204&147\end{bmatrix}$, $\begin{bmatrix}65&236\\168&5\end{bmatrix}$, $\begin{bmatrix}135&188\\69&215\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 124.12.0.g.1 for the level structure with $-I$) |
| Cyclic 248-isogeny field degree: | $64$ |
| Cyclic 248-torsion field degree: | $7680$ |
| Full 248-torsion field degree: | $57139200$ |
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 |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 248.12.0-4.c.1.4 | $248$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 248.48.0-248.be.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.be.1.8 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bf.1.10 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bf.1.12 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bm.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bm.1.8 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bn.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bn.1.8 | $248$ | $2$ | $2$ | $0$ |