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Properties

Label 48.48.1.bg.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: $64$
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: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.411

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}13&33\\4&13\end{bmatrix}$, $\begin{bmatrix}31&14\\4&33\end{bmatrix}$, $\begin{bmatrix}39&8\\16&43\end{bmatrix}$, $\begin{bmatrix}41&3\\24&19\end{bmatrix}$, $\begin{bmatrix}47&40\\28&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bg.1.1, 48.96.1-48.bg.1.2, 48.96.1-48.bg.1.3, 48.96.1-48.bg.1.4, 48.96.1-48.bg.1.5, 48.96.1-48.bg.1.6, 48.96.1-48.bg.1.7, 48.96.1-48.bg.1.8, 48.96.1-48.bg.1.9, 48.96.1-48.bg.1.10, 48.96.1-48.bg.1.11, 48.96.1-48.bg.1.12, 48.96.1-48.bg.1.13, 48.96.1-48.bg.1.14, 48.96.1-48.bg.1.15, 48.96.1-48.bg.1.16, 240.96.1-48.bg.1.1, 240.96.1-48.bg.1.2, 240.96.1-48.bg.1.3, 240.96.1-48.bg.1.4, 240.96.1-48.bg.1.5, 240.96.1-48.bg.1.6, 240.96.1-48.bg.1.7, 240.96.1-48.bg.1.8, 240.96.1-48.bg.1.9, 240.96.1-48.bg.1.10, 240.96.1-48.bg.1.11, 240.96.1-48.bg.1.12, 240.96.1-48.bg.1.13, 240.96.1-48.bg.1.14, 240.96.1-48.bg.1.15, 240.96.1-48.bg.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

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 $ $=$ $ 12 x y - 3 y^{2} + w^{2} $
$=$ $24 x^{2} + 6 x y + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} + 6 x^{2} y^{2} - 9 x^{2} z^{2} + 9 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2\,\frac{6048y^{2}z^{10}-432y^{2}z^{8}w^{2}-340848y^{2}z^{6}w^{4}+3013416y^{2}z^{4}w^{6}-4719654y^{2}z^{2}w^{8}+786429y^{2}w^{10}+2048z^{12}-6144z^{10}w^{2}-18960z^{8}w^{4}+23968z^{6}w^{6}+43392z^{4}w^{8}+524640z^{2}w^{10}-131071w^{12}}{w^{2}z^{2}(24y^{2}z^{6}-132y^{2}z^{4}w^{2}+42y^{2}z^{2}w^{4}-3y^{2}w^{6}-64z^{6}w^{2}+68z^{4}w^{4}-16z^{2}w^{6}+w^{8})}$

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
16.24.1.a.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.by.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.e.2 $48$ $2$ $2$ $0$ $0$ full Jacobian

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.a.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.s.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bh.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.br.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ch.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ck.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cw.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.db.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.ek.2 $48$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.9.bat.2 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.96.1.hp.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ht.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.if.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ij.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ix.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.jf.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.kd.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.kl.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.cm.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.egs.2 $240$ $6$ $6$ $17$ $?$ not computed