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Properties

Label 48.48.1.cs.1
Level $48$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.302

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&2\\40&39\end{bmatrix}$, $\begin{bmatrix}29&34\\16&39\end{bmatrix}$, $\begin{bmatrix}33&28\\32&5\end{bmatrix}$, $\begin{bmatrix}33&43\\20&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.cs.1.1, 48.96.1-48.cs.1.2, 48.96.1-48.cs.1.3, 48.96.1-48.cs.1.4, 48.96.1-48.cs.1.5, 48.96.1-48.cs.1.6, 48.96.1-48.cs.1.7, 48.96.1-48.cs.1.8, 96.96.1-48.cs.1.1, 96.96.1-48.cs.1.2, 96.96.1-48.cs.1.3, 96.96.1-48.cs.1.4, 240.96.1-48.cs.1.1, 240.96.1-48.cs.1.2, 240.96.1-48.cs.1.3, 240.96.1-48.cs.1.4, 240.96.1-48.cs.1.5, 240.96.1-48.cs.1.6, 240.96.1-48.cs.1.7, 240.96.1-48.cs.1.8
Cyclic 48-isogeny field degree: $16$
Cyclic 48-torsion field degree: $256$
Full 48-torsion field degree: $24576$

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 $ $=$ $ - x w + 2 y^{2} $
$=$ $12 x^{2} - z^{2} + 3 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 3 y^{2} z^{2} + 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3}\cdot\frac{(z^{4}+42z^{2}w^{2}+9w^{4})^{3}}{w^{2}z^{2}(z^{2}-3w^{2})^{4}}$

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
16.24.0.j.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.bs.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.c.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
48.144.9.mg.1 $48$ $3$ $3$ $9$ $2$ $1^{8}$
48.192.9.bgx.1 $48$ $4$ $4$ $9$ $2$ $1^{8}$
96.96.3.x.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.x.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.cp.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.cp.2 $96$ $2$ $2$ $3$ $?$ not computed
240.240.17.ho.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.kcw.1 $240$ $6$ $6$ $17$ $?$ not computed