Invariants
Level: | $248$ | $\SL_2$-level: | $4$ | ||||
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: | $\begin{bmatrix}111&124\\132&139\end{bmatrix}$, $\begin{bmatrix}165&210\\214&205\end{bmatrix}$, $\begin{bmatrix}191&20\\94&131\end{bmatrix}$, $\begin{bmatrix}235&16\\22&13\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 248.12.0.b.1 for the level structure with $-I$) |
Cyclic 248-isogeny field degree: | $128$ |
Cyclic 248-torsion field degree: | $15360$ |
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 |
---|---|---|---|---|---|
8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
124.12.0-2.a.1.1 | $124$ | $2$ | $2$ | $0$ | $?$ |
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.c.1.2 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.c.1.3 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.d.1.1 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.d.1.7 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.e.1.7 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.e.1.9 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.f.1.2 | $248$ | $2$ | $2$ | $0$ |
248.48.0-248.f.1.7 | $248$ | $2$ | $2$ | $0$ |