Invariants
Level: | $248$ | $\SL_2$-level: | $4$ | ||||
Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
Cusps: | $3$ (all of which are rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $3$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4B0 |
Level structure
$\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}12&185\\49&188\end{bmatrix}$, $\begin{bmatrix}19&66\\210&19\end{bmatrix}$, $\begin{bmatrix}35&12\\46&53\end{bmatrix}$, $\begin{bmatrix}100&3\\167&212\end{bmatrix}$, $\begin{bmatrix}247&218\\120&181\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
Cyclic 248-isogeny field degree: | $64$ |
Cyclic 248-torsion field degree: | $7680$ |
Full 248-torsion field degree: | $114278400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
248.24.0-4.b.1.11 | $248$ | $2$ | $2$ | $0$ |
248.24.0-4.d.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.d.1.2 | $248$ | $2$ | $2$ | $0$ |
248.24.0-124.g.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-124.h.1.2 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.k.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.m.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.m.1.8 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.n.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.n.1.12 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.o.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.o.1.8 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.p.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-8.p.1.8 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.s.1.3 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.v.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.y.1.3 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.y.1.14 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.z.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.z.1.16 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.ba.1.1 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.ba.1.16 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.bb.1.3 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.bb.1.14 | $248$ | $2$ | $2$ | $0$ |
248.384.14-124.c.1.7 | $248$ | $32$ | $32$ | $14$ |