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Properties

Label 48.192.7.gv.2
Level $48$
Index $192$
Genus $7$
Analytic rank $0$
Cusps $20$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $288$
Index: $192$ $\PSL_2$-index:$192$
Genus: $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$
Cusps: $20$ (none of which are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ Cusp orbits $2^{6}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 48AP7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.7.1220

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&31\\0&43\end{bmatrix}$, $\begin{bmatrix}25&24\\36&19\end{bmatrix}$, $\begin{bmatrix}31&3\\24&37\end{bmatrix}$, $\begin{bmatrix}31&15\\36&23\end{bmatrix}$, $\begin{bmatrix}43&34\\36&1\end{bmatrix}$, $\begin{bmatrix}47&22\\12&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.384.7-48.gv.2.1, 48.384.7-48.gv.2.2, 48.384.7-48.gv.2.3, 48.384.7-48.gv.2.4, 48.384.7-48.gv.2.5, 48.384.7-48.gv.2.6, 48.384.7-48.gv.2.7, 48.384.7-48.gv.2.8, 48.384.7-48.gv.2.9, 48.384.7-48.gv.2.10, 48.384.7-48.gv.2.11, 48.384.7-48.gv.2.12, 48.384.7-48.gv.2.13, 48.384.7-48.gv.2.14, 48.384.7-48.gv.2.15, 48.384.7-48.gv.2.16, 48.384.7-48.gv.2.17, 48.384.7-48.gv.2.18, 48.384.7-48.gv.2.19, 48.384.7-48.gv.2.20, 48.384.7-48.gv.2.21, 48.384.7-48.gv.2.22, 48.384.7-48.gv.2.23, 48.384.7-48.gv.2.24, 48.384.7-48.gv.2.25, 48.384.7-48.gv.2.26, 48.384.7-48.gv.2.27, 48.384.7-48.gv.2.28, 48.384.7-48.gv.2.29, 48.384.7-48.gv.2.30, 48.384.7-48.gv.2.31, 48.384.7-48.gv.2.32, 240.384.7-48.gv.2.1, 240.384.7-48.gv.2.2, 240.384.7-48.gv.2.3, 240.384.7-48.gv.2.4, 240.384.7-48.gv.2.5, 240.384.7-48.gv.2.6, 240.384.7-48.gv.2.7, 240.384.7-48.gv.2.8, 240.384.7-48.gv.2.9, 240.384.7-48.gv.2.10, 240.384.7-48.gv.2.11, 240.384.7-48.gv.2.12, 240.384.7-48.gv.2.13, 240.384.7-48.gv.2.14, 240.384.7-48.gv.2.15, 240.384.7-48.gv.2.16, 240.384.7-48.gv.2.17, 240.384.7-48.gv.2.18, 240.384.7-48.gv.2.19, 240.384.7-48.gv.2.20, 240.384.7-48.gv.2.21, 240.384.7-48.gv.2.22, 240.384.7-48.gv.2.23, 240.384.7-48.gv.2.24, 240.384.7-48.gv.2.25, 240.384.7-48.gv.2.26, 240.384.7-48.gv.2.27, 240.384.7-48.gv.2.28, 240.384.7-48.gv.2.29, 240.384.7-48.gv.2.30, 240.384.7-48.gv.2.31, 240.384.7-48.gv.2.32
Cyclic 48-isogeny field degree: $2$
Cyclic 48-torsion field degree: $32$
Full 48-torsion field degree: $6144$

Jacobian

Conductor: $2^{30}\cdot3^{11}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2^{2}$
Newforms: 24.2.a.a, 72.2.a.a, 96.2.d.a, 144.2.a.b, 288.2.d.b

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ x t - y z + y v $
$=$ $y z + y v - w u$
$=$ $2 y w - t v + u v$
$=$ $2 x y + z t + z u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 6 x^{6} y^{2} - x^{6} z^{2} - 18 x^{4} y^{4} + 21 x^{4} y^{2} z^{2} - 2 x^{4} z^{4} + 9 x^{2} y^{4} z^{2} + \cdots + 27 y^{6} z^{2} $
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Rational points

This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.gg.1 :

$\displaystyle X$ $=$ $\displaystyle -y$
$\displaystyle Y$ $=$ $\displaystyle 2y+z-v$
$\displaystyle Z$ $=$ $\displaystyle 2y-v$

Equation of the image curve:

$0$ $=$ $ 6X^{4}-8X^{3}Y-3X^{2}Y^{2}+2XY^{3}-4X^{3}Z-6X^{2}YZ-6XY^{2}Z+Y^{3}Z+6X^{2}Z^{2}-3Y^{2}Z^{2}+4XZ^{3}+2YZ^{3} $

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
24.96.3.gg.1 $24$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$
48.48.0.bi.2 $48$ $4$ $4$ $0$ $0$ full Jacobian
48.96.3.qd.1 $48$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$
48.96.3.qf.1 $48$ $2$ $2$ $3$ $0$ $2^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.384.13.ln.3 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.ln.4 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.lv.1 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.lv.2 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.mu.3 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.mu.4 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.nc.1 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.13.nc.2 $48$ $2$ $2$ $13$ $0$ $2^{3}$
48.384.17.in.1 $48$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
48.384.17.jz.1 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.pu.2 $48$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
48.384.17.qf.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.wj.1 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.wr.1 $48$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
48.384.17.xo.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.xw.2 $48$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
48.384.17.bny.1 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bny.2 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bog.3 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bog.4 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bpd.1 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bpd.2 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bpl.3 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bpl.4 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.576.29.hm.1 $48$ $3$ $3$ $29$ $0$ $1^{10}\cdot2^{6}$
240.384.13.gav.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gav.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gbd.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gbd.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gcc.3 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gcc.4 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gck.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.gck.4 $240$ $2$ $2$ $13$ $?$ not computed
240.384.17.ldz.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.leh.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.lfn.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.lfv.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.lhr.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.lhz.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.ljf.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.ljn.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.msa.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.msa.4 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.msi.3 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.msi.4 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.mtf.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.mtf.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.mtn.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.mtn.2 $240$ $2$ $2$ $17$ $?$ not computed