Modular curve 48.192.3-48.co.2.1

• Label: 48.192.3-48.co.2.1
• Level: $48$
• Index: $192$
• Genus: $3$
• Analytic rank: $0$
• Cusps: $12$
• $\Q$-cusps: $0$

Invariants

Level:	$48$		$\SL_2$-level:	$16$		Newform level:	$576$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)		Cusp widths	$4^{8}\cdot16^{4}$		Cusp orbits	$2^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	16J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	48.192.3.1246

Level structure

$\GL_2(\Z/48\Z)$-generators:	$\begin{bmatrix}19&2\\16&9\end{bmatrix}$, $\begin{bmatrix}21&38\\16&17\end{bmatrix}$, $\begin{bmatrix}47&2\\8&25\end{bmatrix}$, $\begin{bmatrix}47&28\\40&21\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 48.96.3.co.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree:	$8$
Cyclic 48-torsion field degree:	$64$
Full 48-torsion field degree:	$6144$

Jacobian

Conductor:	$2^{17}\cdot3^{4}$
Simple:	no
Squarefree:	yes
Decomposition:	$1\cdot2$
Newforms:	32.2.a.a, 576.2.d.a

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

$ 0 $	$=$	$ x t - x u + w t + w u $
	$=$	$2 x t + z u$
	$=$	$2 x^{2} + x z - 2 x w + z w$
	$=$	$3 x z + y^{2} + z w$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 900 x^{4} y^{2} + 600 x^{4} z^{2} - 144 x^{2} y^{4} - 336 x^{2} y^{2} z^{2} + 240 x^{2} z^{4} + \cdots + 24 z^{6} $

Geometric Weierstrass model Geometric Weierstrass model

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

Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{2^8}{3\cdot5^4}\cdot\frac{26562549252096zw^{11}+40800963133440zw^{9}u^{2}+30984839331840zw^{7}u^{4}+3357793751040zw^{5}u^{6}-7993355823360zw^{3}u^{8}-3907359461760zwu^{10}-47010758787072w^{12}-41195686330368w^{10}u^{2}+5400964251648w^{8}u^{4}+21450595903488w^{6}u^{6}+12075698265408w^{4}u^{8}+1648415282928w^{2}u^{10}-3955078125t^{12}-47460937500t^{11}u-237304687500t^{10}u^{2}-632812500000t^{9}u^{3}-961083984375t^{8}u^{4}-854296875000t^{7}u^{5}-514160156250t^{6}u^{6}-237304687500t^{5}u^{7}+23730468750t^{4}u^{8}-94921875000t^{3}u^{9}-94921875000t^{2}u^{10}+82137899608tu^{11}+29231784767u^{12}}{u^{4}(2006581248zw^{7}-54743040zw^{5}u^{2}-826087680zw^{3}u^{4}-258602880zwu^{6}+2974040064w^{8}+3881281536w^{6}u^{2}+1622147904w^{4}u^{4}+252830064w^{2}u^{6}-2109375t^{4}u^{4}-8437500t^{3}u^{5}-1826896tu^{7}+4597771u^{8})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.co.2 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle 4w$
$\displaystyle Z$	$=$	$\displaystyle u$

Equation of the image curve:

$0$

$=$

$ 900X^{4}Y^{2}-144X^{2}Y^{4}+9Y^{6}+600X^{4}Z^{2}-336X^{2}Y^{2}Z^{2}+54Y^{4}Z^{2}+240X^{2}Z^{4}+89Y^{2}Z^{4}+24Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
16.96.1-16.b.2.2	$16$	$2$	$2$	$1$	$0$	$2$
24.96.0-24.bc.1.3	$24$	$2$	$2$	$0$	$0$	full Jacobian
48.96.0-24.bc.1.3	$48$	$2$	$2$	$0$	$0$	full Jacobian
48.96.1-16.b.2.11	$48$	$2$	$2$	$1$	$0$	$2$
48.96.2-48.d.2.4	$48$	$2$	$2$	$2$	$0$	$1$
48.96.2-48.d.2.5	$48$	$2$	$2$	$2$	$0$	$1$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.384.5-48.bg.2.2	$48$	$2$	$2$	$5$	$0$	$1^{2}$
48.384.5-48.cn.1.2	$48$	$2$	$2$	$5$	$0$	$1^{2}$
48.384.5-48.ej.1.1	$48$	$2$	$2$	$5$	$1$	$1^{2}$
48.384.5-48.en.2.1	$48$	$2$	$2$	$5$	$1$	$1^{2}$
48.576.19-48.mz.1.9	$48$	$3$	$3$	$19$	$1$	$1^{8}\cdot2^{2}\cdot4$
48.768.21-48.kb.2.2	$48$	$4$	$4$	$21$	$0$	$1^{8}\cdot2^{3}\cdot4$
240.384.5-240.tw.1.4	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.ui.1.4	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.ye.1.2	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.yq.2.3	$240$	$2$	$2$	$5$	$?$	not computed