Modular curve 48.96.1.by.1

• Label: 48.96.1.by.1
• Level: $48$
• Index: $96$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$48$		$\SL_2$-level:	$16$		Newform level:	$288$
Index:	$96$		$\PSL_2$-index:	$96$
Genus:	$1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$2^{8}\cdot4^{4}\cdot16^{4}$		Cusp orbits	$2^{4}\cdot4^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	48.96.1.234

Level structure

$\GL_2(\Z/48\Z)$-generators:	$\begin{bmatrix}7&12\\4&17\end{bmatrix}$, $\begin{bmatrix}37&7\\4&5\end{bmatrix}$, $\begin{bmatrix}37&27\\0&7\end{bmatrix}$, $\begin{bmatrix}39&19\\20&47\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	48.192.1-48.by.1.1, 48.192.1-48.by.1.2, 48.192.1-48.by.1.3, 48.192.1-48.by.1.4, 48.192.1-48.by.1.5, 48.192.1-48.by.1.6, 48.192.1-48.by.1.7, 48.192.1-48.by.1.8, 240.192.1-48.by.1.1, 240.192.1-48.by.1.2, 240.192.1-48.by.1.3, 240.192.1-48.by.1.4, 240.192.1-48.by.1.5, 240.192.1-48.by.1.6, 240.192.1-48.by.1.7, 240.192.1-48.by.1.8
Cyclic 48-isogeny field degree:	$8$
Cyclic 48-torsion field degree:	$64$
Full 48-torsion field degree:	$12288$

Jacobian

Conductor:	$2^{5}\cdot3^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $	$=$	$ 2 x^{2} + 3 y^{2} + 3 y z $
	$=$	$4 x^{2} - 2 y^{2} - 2 y z - z^{2} + 2 w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 9 x^{4} - 36 x^{2} z^{2} - 2 y^{2} z^{2} + 4 z^{4} $

Rational points

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

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle 3w$
$\displaystyle Z$	$=$	$\displaystyle \frac{3}{2}y$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2\,\frac{(z^{8}+120z^{6}w^{2}+536z^{4}w^{4}+480z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{2}(z^{2}-2w^{2})^{8}(z^{2}+2w^{2})^{2}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.48.0.p.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
48.48.0.l.1	$48$	$2$	$2$	$0$	$0$	full Jacobian
48.48.0.bn.1	$48$	$2$	$2$	$0$	$0$	full Jacobian
48.48.0.bp.1	$48$	$2$	$2$	$0$	$0$	full Jacobian
48.48.1.j.1	$48$	$2$	$2$	$1$	$0$	dimension zero
48.48.1.bz.1	$48$	$2$	$2$	$1$	$0$	dimension zero
48.48.1.cb.2	$48$	$2$	$2$	$1$	$0$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.288.17.qj.2	$48$	$3$	$3$	$17$	$3$	$1^{8}\cdot2^{4}$
48.384.17.qp.2	$48$	$4$	$4$	$17$	$2$	$1^{8}\cdot2^{4}$