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Properties

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

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
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 $2^{4}$
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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.83

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&14\\16&27\end{bmatrix}$, $\begin{bmatrix}11&10\\24&43\end{bmatrix}$, $\begin{bmatrix}31&46\\16&47\end{bmatrix}$, $\begin{bmatrix}41&23\\0&43\end{bmatrix}$, $\begin{bmatrix}41&43\\40&37\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.j.1.1, 48.96.1-48.j.1.2, 48.96.1-48.j.1.3, 48.96.1-48.j.1.4, 48.96.1-48.j.1.5, 48.96.1-48.j.1.6, 48.96.1-48.j.1.7, 48.96.1-48.j.1.8, 48.96.1-48.j.1.9, 48.96.1-48.j.1.10, 48.96.1-48.j.1.11, 48.96.1-48.j.1.12, 48.96.1-48.j.1.13, 48.96.1-48.j.1.14, 48.96.1-48.j.1.15, 48.96.1-48.j.1.16, 240.96.1-48.j.1.1, 240.96.1-48.j.1.2, 240.96.1-48.j.1.3, 240.96.1-48.j.1.4, 240.96.1-48.j.1.5, 240.96.1-48.j.1.6, 240.96.1-48.j.1.7, 240.96.1-48.j.1.8, 240.96.1-48.j.1.9, 240.96.1-48.j.1.10, 240.96.1-48.j.1.11, 240.96.1-48.j.1.12, 240.96.1-48.j.1.13, 240.96.1-48.j.1.14, 240.96.1-48.j.1.15, 240.96.1-48.j.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

$ 0 $ $=$ $ 18 x^{2} - 6 x y - w^{2} $
$=$ $14 x^{2} + 12 x y - 2 y^{2} + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} + 2 x^{2} y^{2} - 4 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{8}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{4}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{5}{3^2}\cdot\frac{1194102y^{2}z^{10}+2146176y^{2}z^{6}w^{4}+41472y^{2}z^{2}w^{8}+336069z^{12}+1194102z^{10}w^{2}+2245077z^{8}w^{4}+2146176z^{6}w^{6}+499824z^{4}w^{8}+41472z^{2}w^{10}+1280w^{12}}{w^{4}z^{2}(18y^{2}z^{4}+32y^{2}w^{4}-9z^{6}+18z^{4}w^{2}-41z^{2}w^{4}+32w^{6})}$

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
8.24.0.r.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.h.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.b.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
48.96.1.bx.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bx.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.by.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.by.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bz.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bz.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ca.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ca.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.bm.1 $48$ $3$ $3$ $9$ $3$ $1^{8}$
48.192.9.mq.1 $48$ $4$ $4$ $9$ $2$ $1^{8}$
240.96.1.gb.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gb.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gc.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gc.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gd.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gd.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ge.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ge.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.t.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.bfx.1 $240$ $6$ $6$ $17$ $?$ not computed