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Properties

Label 48.96.1.t.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: 48.96.1.2140

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&5\\4&11\end{bmatrix}$, $\begin{bmatrix}15&5\\4&15\end{bmatrix}$, $\begin{bmatrix}31&10\\36&5\end{bmatrix}$, $\begin{bmatrix}43&4\\0&35\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.t.2.1, 48.192.1-48.t.2.2, 48.192.1-48.t.2.3, 48.192.1-48.t.2.4, 48.192.1-48.t.2.5, 48.192.1-48.t.2.6, 48.192.1-48.t.2.7, 48.192.1-48.t.2.8, 96.192.1-48.t.2.1, 96.192.1-48.t.2.2, 96.192.1-48.t.2.3, 96.192.1-48.t.2.4, 240.192.1-48.t.2.1, 240.192.1-48.t.2.2, 240.192.1-48.t.2.3, 240.192.1-48.t.2.4, 240.192.1-48.t.2.5, 240.192.1-48.t.2.6, 240.192.1-48.t.2.7, 240.192.1-48.t.2.8
Cyclic 48-isogeny field degree: $4$
Cyclic 48-torsion field degree: $64$
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^{2} - y^{2} - y z - z^{2} $
$=$ $6 x^{2} + 6 y^{2} + 6 y z + 12 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 10 x^{2} y^{2} + 6 x^{2} z^{2} + 49 y^{4} - 84 y^{2} z^{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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^2}{3^2}\cdot\frac{(1296z^{8}+6048z^{6}w^{2}-360z^{4}w^{4}-24z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{4}(6z^{2}-w^{2})^{8}(12z^{2}-w^{2})}$

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.d.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bd.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.f.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bu.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bv.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bi.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bj.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.fq.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.jn.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.m.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.p.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.v.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.y.2 $96$ $2$ $2$ $5$ $?$ not computed