Modular curve 248.12.0.v.1

• Label: 248.12.0.v.1
• Level: $248$
• Index: $12$
• Genus: $0$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$248$		$\SL_2$-level:	$4$
Index:	$12$		$\PSL_2$-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:	$\begin{bmatrix}9&44\\86&207\end{bmatrix}$, $\begin{bmatrix}27&8\\89&153\end{bmatrix}$, $\begin{bmatrix}79&24\\190&55\end{bmatrix}$, $\begin{bmatrix}109&72\\170&243\end{bmatrix}$, $\begin{bmatrix}165&0\\219&213\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	248.24.0-248.v.1.1, 248.24.0-248.v.1.2, 248.24.0-248.v.1.3, 248.24.0-248.v.1.4, 248.24.0-248.v.1.5, 248.24.0-248.v.1.6, 248.24.0-248.v.1.7, 248.24.0-248.v.1.8
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 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
$X_0(4)$	$4$	$2$	$2$	$0$	$0$
248.6.0.b.1	$248$	$2$	$2$	$0$	$?$
248.6.0.f.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
248.24.0.bk.1	$248$	$2$	$2$	$0$
248.24.0.bl.1	$248$	$2$	$2$	$0$
248.24.0.bs.1	$248$	$2$	$2$	$0$
248.24.0.bt.1	$248$	$2$	$2$	$0$