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Properties

Label 48.96.1.cb.2
Level $48$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $32$
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 $4^{4}$
Elliptic points: $0$ of order $2$ and $0$ 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: 16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.2176

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&41\\28&33\end{bmatrix}$, $\begin{bmatrix}15&47\\4&23\end{bmatrix}$, $\begin{bmatrix}17&22\\40&9\end{bmatrix}$, $\begin{bmatrix}37&10\\36&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.cb.2.1, 48.192.1-48.cb.2.2, 48.192.1-48.cb.2.3, 48.192.1-48.cb.2.4, 48.192.1-48.cb.2.5, 48.192.1-48.cb.2.6, 48.192.1-48.cb.2.7, 48.192.1-48.cb.2.8, 96.192.1-48.cb.2.1, 96.192.1-48.cb.2.2, 96.192.1-48.cb.2.3, 96.192.1-48.cb.2.4, 240.192.1-48.cb.2.1, 240.192.1-48.cb.2.2, 240.192.1-48.cb.2.3, 240.192.1-48.cb.2.4, 240.192.1-48.cb.2.5, 240.192.1-48.cb.2.6, 240.192.1-48.cb.2.7, 240.192.1-48.cb.2.8
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

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} - z^{2} $
$=$ $9 x^{2} - 6 y^{2} - z^{2} - w^{2}$
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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 to other modular curves

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

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

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
16.48.1.j.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bf.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.l.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bw.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.by.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bk.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bm.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.qq.2 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.qs.1 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
96.192.9.dv.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.dx.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.eb.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.ef.1 $96$ $2$ $2$ $9$ $?$ not computed