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Properties

Label 48.96.1.cs.1
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: $32$
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 $2^{4}\cdot4^{2}$
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: 48.96.1.1702

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}15&1\\44&15\end{bmatrix}$, $\begin{bmatrix}17&32\\40&29\end{bmatrix}$, $\begin{bmatrix}17&35\\8&27\end{bmatrix}$, $\begin{bmatrix}43&27\\4&43\end{bmatrix}$, $\begin{bmatrix}47&4\\16&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.cs.1.1, 48.192.1-48.cs.1.2, 48.192.1-48.cs.1.3, 48.192.1-48.cs.1.4, 48.192.1-48.cs.1.5, 48.192.1-48.cs.1.6, 48.192.1-48.cs.1.7, 48.192.1-48.cs.1.8, 48.192.1-48.cs.1.9, 48.192.1-48.cs.1.10, 48.192.1-48.cs.1.11, 48.192.1-48.cs.1.12, 96.192.1-48.cs.1.1, 96.192.1-48.cs.1.2, 96.192.1-48.cs.1.3, 96.192.1-48.cs.1.4, 96.192.1-48.cs.1.5, 96.192.1-48.cs.1.6, 96.192.1-48.cs.1.7, 96.192.1-48.cs.1.8, 240.192.1-48.cs.1.1, 240.192.1-48.cs.1.2, 240.192.1-48.cs.1.3, 240.192.1-48.cs.1.4, 240.192.1-48.cs.1.5, 240.192.1-48.cs.1.6, 240.192.1-48.cs.1.7, 240.192.1-48.cs.1.8, 240.192.1-48.cs.1.9, 240.192.1-48.cs.1.10, 240.192.1-48.cs.1.11, 240.192.1-48.cs.1.12
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ 6 x^{2} - y w - w^{2} $
$=$ $y^{2} + 8 y w - 6 z^{2} + 8 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 36 x^{2} z^{2} - 6 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3}\cdot\frac{4251528yz^{22}w+268791048yz^{20}w^{3}+4944841992yz^{18}w^{5}+41071886244yz^{16}w^{7}+183263146704yz^{14}w^{9}+474733116432yz^{12}w^{11}+730998404784yz^{10}w^{13}+654785649288yz^{8}w^{15}+316037513160yz^{6}w^{17}+70831627848yz^{4}w^{19}+7203397032yz^{2}w^{21}+271669860yw^{23}-531441z^{24}-85030560z^{22}w^{2}-2165917320z^{20}w^{4}-20333405052z^{18}w^{6}-88587061587z^{16}w^{8}-182459485440z^{14}w^{10}-110159341632z^{12}w^{12}+219728793960z^{10}w^{14}+445479541101z^{8}w^{16}+296544997728z^{6}w^{18}+75398085576z^{4}w^{20}+8151155172z^{2}w^{22}+318281039w^{24}}{w^{8}z^{2}(233280yz^{12}w+10023264yz^{10}w^{3}+120000528yz^{8}w^{5}+605560968yz^{6}w^{7}+1468002888yz^{4}w^{9}+1693055208yz^{2}w^{11}+745778864yw^{13}-34992z^{14}-3709152z^{12}w^{2}-59087880z^{10}w^{4}-309950712z^{8}w^{6}-566507547z^{6}w^{8}+72432000z^{4}w^{10}+1192519620z^{2}w^{12}+873734288w^{14})}$

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.v.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bl.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.w.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bk.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bn.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.t.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bk.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
48.192.5.iy.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.iz.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ja.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.jb.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.bat.1 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.wv.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$
96.192.5.ef.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.eg.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.en.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.eo.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.gk.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.gs.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.jm.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.ju.2 $96$ $2$ $2$ $9$ $?$ not computed
240.192.5.ccm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ccn.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cco.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ccp.2 $240$ $2$ $2$ $5$ $?$ not computed