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Properties

Label 48.48.1.w.1
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{4}$
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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.446

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&14\\24&35\end{bmatrix}$, $\begin{bmatrix}13&17\\44&21\end{bmatrix}$, $\begin{bmatrix}25&35\\32&35\end{bmatrix}$, $\begin{bmatrix}37&47\\20&45\end{bmatrix}$, $\begin{bmatrix}45&22\\4&39\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.w.1.1, 48.96.1-48.w.1.2, 48.96.1-48.w.1.3, 48.96.1-48.w.1.4, 48.96.1-48.w.1.5, 48.96.1-48.w.1.6, 48.96.1-48.w.1.7, 48.96.1-48.w.1.8, 48.96.1-48.w.1.9, 48.96.1-48.w.1.10, 48.96.1-48.w.1.11, 48.96.1-48.w.1.12, 48.96.1-48.w.1.13, 48.96.1-48.w.1.14, 48.96.1-48.w.1.15, 48.96.1-48.w.1.16, 96.96.1-48.w.1.1, 96.96.1-48.w.1.2, 96.96.1-48.w.1.3, 96.96.1-48.w.1.4, 96.96.1-48.w.1.5, 96.96.1-48.w.1.6, 96.96.1-48.w.1.7, 96.96.1-48.w.1.8, 240.96.1-48.w.1.1, 240.96.1-48.w.1.2, 240.96.1-48.w.1.3, 240.96.1-48.w.1.4, 240.96.1-48.w.1.5, 240.96.1-48.w.1.6, 240.96.1-48.w.1.7, 240.96.1-48.w.1.8, 240.96.1-48.w.1.9, 240.96.1-48.w.1.10, 240.96.1-48.w.1.11, 240.96.1-48.w.1.12, 240.96.1-48.w.1.13, 240.96.1-48.w.1.14, 240.96.1-48.w.1.15, 240.96.1-48.w.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

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 z - w^{2} $
$=$ $48 x^{2} + y^{2} - 3 z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} - 3 x^{2} y^{2} - z^{4} $
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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:

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

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{y^{12}-672y^{8}w^{4}+162048y^{4}w^{8}+2985255z^{12}-15863040z^{8}w^{4}+28200960z^{4}w^{8}-16769024w^{12}}{w^{4}(y^{8}+48y^{4}w^{4}-81z^{8}+144z^{4}w^{4})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.24.1.b.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.bl.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.g.1 $48$ $2$ $2$ $0$ $0$ 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
48.96.1.dd.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dd.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.de.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.de.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.df.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.df.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dg.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.cs.1 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.zv.1 $48$ $4$ $4$ $9$ $0$ $1^{8}$
96.96.5.t.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.t.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bl.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bl.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bn.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bn.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.ch.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.ch.2 $96$ $2$ $2$ $5$ $?$ not computed
240.96.1.hh.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hh.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hi.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hi.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hj.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hj.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hk.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.hk.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.bm.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.cym.1 $240$ $6$ $6$ $17$ $?$ not computed