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Properties

Label 48.96.1.dv.2
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: 48.96.1.1026

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&14\\4&47\end{bmatrix}$, $\begin{bmatrix}25&47\\20&33\end{bmatrix}$, $\begin{bmatrix}27&5\\8&33\end{bmatrix}$, $\begin{bmatrix}31&8\\20&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.dv.2.1, 48.192.1-48.dv.2.2, 48.192.1-48.dv.2.3, 48.192.1-48.dv.2.4, 48.192.1-48.dv.2.5, 48.192.1-48.dv.2.6, 48.192.1-48.dv.2.7, 48.192.1-48.dv.2.8, 240.192.1-48.dv.2.1, 240.192.1-48.dv.2.2, 240.192.1-48.dv.2.3, 240.192.1-48.dv.2.4, 240.192.1-48.dv.2.5, 240.192.1-48.dv.2.6, 240.192.1-48.dv.2.7, 240.192.1-48.dv.2.8
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 $ $=$ $ 2 x^{2} - 2 y^{2} - z w + w^{2} $
$=$ $4 x^{2} + 2 y^{2} + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} + 8 y^{2} z^{2} + 4 z^{4} $
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Rational points

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

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3}\cdot\frac{(z^{4}+4z^{3}w-12z^{2}w^{2}+4zw^{3}+w^{4})^{3}(2z^{4}-4z^{3}w-6z^{2}w^{2}+8zw^{3}-w^{4})^{3}}{z^{2}(z-2w)^{2}(z^{2}-zw+w^{2})^{8}(z^{2}+2zw-2w^{2})^{2}}$

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
16.48.0.ba.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bi.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.v.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bw.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bq.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bu.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.cg.1 $48$ $2$ $2$ $1$ $1$ 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.cko.2 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.384.17.bll.1 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$