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Properties

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

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
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: $1$
$\Q$-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: 48.96.1.276

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&10\\36&7\end{bmatrix}$, $\begin{bmatrix}9&35\\20&9\end{bmatrix}$, $\begin{bmatrix}17&37\\40&43\end{bmatrix}$, $\begin{bmatrix}43&9\\24&41\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.bt.1.1, 48.192.1-48.bt.1.2, 48.192.1-48.bt.1.3, 48.192.1-48.bt.1.4, 48.192.1-48.bt.1.5, 48.192.1-48.bt.1.6, 48.192.1-48.bt.1.7, 48.192.1-48.bt.1.8, 240.192.1-48.bt.1.1, 240.192.1-48.bt.1.2, 240.192.1-48.bt.1.3, 240.192.1-48.bt.1.4, 240.192.1-48.bt.1.5, 240.192.1-48.bt.1.6, 240.192.1-48.bt.1.7, 240.192.1-48.bt.1.8
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

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

$ 0 $ $=$ $ 4 x^{2} + 2 y^{2} + w^{2} $
$=$ $x^{2} - y^{2} - 4 z^{2} + w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^8\cdot3}\cdot\frac{(256z^{8}-3072z^{6}w^{2}+2880z^{4}w^{4}-864z^{2}w^{6}+81w^{8})^{3}}{w^{2}z^{16}(8z^{2}-3w^{2})^{2}(16z^{2}-3w^{2})}$

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
8.48.0.o.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.l.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.be.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bg.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.i.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bq.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bs.2 $48$ $2$ $2$ $1$ $1$ 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.ou.2 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.384.17.qd.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$