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Properties

Label 48.96.1.a.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: $64$
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 $2^{2}\cdot4^{3}$
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: 16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.1441

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&16\\28&15\end{bmatrix}$, $\begin{bmatrix}11&20\\32&31\end{bmatrix}$, $\begin{bmatrix}15&40\\4&25\end{bmatrix}$, $\begin{bmatrix}47&22\\44&33\end{bmatrix}$, $\begin{bmatrix}47&26\\28&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.a.2.1, 48.192.1-48.a.2.2, 48.192.1-48.a.2.3, 48.192.1-48.a.2.4, 48.192.1-48.a.2.5, 48.192.1-48.a.2.6, 48.192.1-48.a.2.7, 48.192.1-48.a.2.8, 48.192.1-48.a.2.9, 48.192.1-48.a.2.10, 48.192.1-48.a.2.11, 48.192.1-48.a.2.12, 240.192.1-48.a.2.1, 240.192.1-48.a.2.2, 240.192.1-48.a.2.3, 240.192.1-48.a.2.4, 240.192.1-48.a.2.5, 240.192.1-48.a.2.6, 240.192.1-48.a.2.7, 240.192.1-48.a.2.8, 240.192.1-48.a.2.9, 240.192.1-48.a.2.10, 240.192.1-48.a.2.11, 240.192.1-48.a.2.12
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 3 x y - 3 y^{2} - 2 z^{2} $
$=$ $6 x^{2} - 3 x y + 3 y^{2} - 6 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 6 x^{2} y^{2} + 36 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^8}\cdot\frac{(256z^{8}-64z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{16}(8z^{2}-w^{2})(8z^{2}+w^{2})}$

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.a.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.ba.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.d.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bw.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.by.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bi.1 $48$ $2$ $2$ $1$ $0$ 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
48.192.5.y.3 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ba.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bb.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bc.2 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.288.17.dg.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.hn.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
240.192.5.bdd.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bde.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdf.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdg.2 $240$ $2$ $2$ $5$ $?$ not computed