Modular curve 48.64.1.b.1

• Label: 48.64.1.b.1
• Level: $48$
• Index: $64$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$48$		$\SL_2$-level:	$8$		Newform level:	$32$
Index:	$64$		$\PSL_2$-index:	$64$
Genus:	$1 = 1 + \frac{ 64 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$8^{8}$		Cusp orbits	$4^{2}$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2 \le \gamma \le 4$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	8J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	48.64.1.1

Level structure

$\GL_2(\Z/48\Z)$-generators:	$\begin{bmatrix}5&45\\9&32\end{bmatrix}$, $\begin{bmatrix}15&46\\43&1\end{bmatrix}$, $\begin{bmatrix}45&47\\35&22\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 48-isogeny field degree:	$48$
Cyclic 48-torsion field degree:	$768$
Full 48-torsion field degree:	$18432$

Jacobian

Conductor:	$2^{5}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	32.2.a.a

Models

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

$ 0 $	$=$	$ 3 x^{2} - 6 y^{2} - 2 z^{2} - 2 z w $
	$=$	$9 x^{2} + 12 x y + 6 y^{2} + 3 z^{2} + 4 z w - w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 81 x^{4} + 180 x^{3} z + 84 x^{2} y^{2} + 82 x^{2} z^{2} + 96 x y^{2} z - 20 x z^{3} + 18 y^{4} + \cdots + 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:

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle 2y$
$\displaystyle Z$	$=$	$\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 64 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^6}\cdot\frac{1848000y^{2}z^{14}+67824000y^{2}z^{13}w+884041920y^{2}z^{12}w^{2}+4449938688y^{2}z^{11}w^{3}+5570276544y^{2}z^{10}w^{4}-2411876736y^{2}z^{9}w^{5}-4614420288y^{2}z^{8}w^{6}+2164962816y^{2}z^{7}w^{7}+1012544064y^{2}z^{6}w^{8}-1062322560y^{2}z^{5}w^{9}+368986176y^{2}z^{4}w^{10}-68408064y^{2}z^{3}w^{11}+7228992y^{2}z^{2}w^{12}-411264y^{2}zw^{13}+9792y^{2}w^{14}+640625z^{16}+23693200z^{15}w+324863240z^{14}w^{2}+1825350288z^{13}w^{3}+3455417980z^{12}w^{4}+1094489232z^{11}w^{5}-2501798408z^{10}w^{6}-868802800z^{9}w^{7}+1188393286z^{8}w^{8}-36821712z^{7}w^{9}-271768648z^{6}w^{10}+128693040z^{5}w^{11}-29396164z^{4}w^{12}+3892912z^{3}w^{13}-304696z^{2}w^{14}+13104zw^{15}-239w^{16}}{z^{8}(10080y^{2}z^{6}+39936y^{2}z^{5}w+42912y^{2}z^{4}w^{2}-2304y^{2}z^{3}w^{3}-13536y^{2}z^{2}w^{4}+3840y^{2}zw^{5}-288y^{2}w^{6}+3431z^{8}+17144z^{7}w+28588z^{6}w^{2}+13880z^{5}w^{3}-6038z^{4}w^{4}-3448z^{3}w^{5}+1516z^{2}w^{6}-184zw^{7}+7w^{8})}$

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
8.32.1.b.1	$8$	$2$	$2$	$1$	$0$	dimension zero
48.16.0.b.1	$48$	$4$	$4$	$0$	$0$	full Jacobian

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.192.5.gf.1	$48$	$3$	$3$	$5$	$2$	$1^{4}$
48.192.13.lo.1	$48$	$3$	$3$	$13$	$5$	$1^{6}\cdot2^{3}$
48.256.13.b.1	$48$	$4$	$4$	$13$	$5$	$1^{10}\cdot2$
48.256.13.f.1	$48$	$4$	$4$	$13$	$5$	$1^{10}\cdot2$
240.320.21.dv.1	$240$	$5$	$5$	$21$	$?$	not computed