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Properties

Label 248.48.0.s.2
Level $248$
Index $48$
Genus $0$
Cusps $10$
$\Q$-cusps $0$

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Invariants

Level: $248$ $\SL_2$-level: $8$
Index: $48$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot8^{4}$ Cusp orbits $2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1 \le \gamma \le 2$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8O0

Level structure

$\GL_2(\Z/248\Z)$-generators: $\begin{bmatrix}15&48\\148&69\end{bmatrix}$, $\begin{bmatrix}117&192\\216&107\end{bmatrix}$, $\begin{bmatrix}173&0\\144&121\end{bmatrix}$, $\begin{bmatrix}243&12\\114&167\end{bmatrix}$, $\begin{bmatrix}243&52\\100&239\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 248.96.0-248.s.2.1, 248.96.0-248.s.2.2, 248.96.0-248.s.2.3, 248.96.0-248.s.2.4, 248.96.0-248.s.2.5, 248.96.0-248.s.2.6, 248.96.0-248.s.2.7, 248.96.0-248.s.2.8, 248.96.0-248.s.2.9, 248.96.0-248.s.2.10, 248.96.0-248.s.2.11, 248.96.0-248.s.2.12, 248.96.0-248.s.2.13, 248.96.0-248.s.2.14, 248.96.0-248.s.2.15, 248.96.0-248.s.2.16
Cyclic 248-isogeny field degree: $64$
Cyclic 248-torsion field degree: $7680$
Full 248-torsion field degree: $28569600$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$
248.24.0.h.2 $248$ $2$ $2$ $0$ $?$
248.24.0.m.1 $248$ $2$ $2$ $0$ $?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
248.96.1.b.1 $248$ $2$ $2$ $1$
248.96.1.c.1 $248$ $2$ $2$ $1$
248.96.1.h.2 $248$ $2$ $2$ $1$
248.96.1.i.2 $248$ $2$ $2$ $1$
248.96.1.bi.2 $248$ $2$ $2$ $1$
248.96.1.bj.2 $248$ $2$ $2$ $1$
248.96.1.bk.1 $248$ $2$ $2$ $1$
248.96.1.bl.1 $248$ $2$ $2$ $1$