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Properties

Label 48.192.1-48.dx.2.7
Level $48$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $192$ $\PSL_2$-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 $2^{2}\cdot4^{3}$
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: 48.192.1.750

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}23&31\\16&25\end{bmatrix}$, $\begin{bmatrix}25&38\\40&13\end{bmatrix}$, $\begin{bmatrix}29&28\\28&3\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.1.dx.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $32$
Full 48-torsion field degree: $6144$

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

$ 0 $ $=$ $ 4 x^{4} + 20 x^{2} y^{2} - 2 x^{2} z^{2} + 49 y^{4} - 14 y^{2} z^{2} + z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{7^2}\cdot\frac{345120060yz^{23}+675972559584yz^{21}w^{2}+93948782137560yz^{19}w^{4}+2757682810005480yz^{17}w^{6}+15636478945448760yz^{15}w^{8}+33092853241892640yz^{13}w^{10}+34031916854774832yz^{11}w^{12}+37573539175755408yz^{9}w^{14}+58541592133508988yz^{7}w^{16}+45238650666361152yz^{5}w^{18}+14997029032778520yz^{3}w^{20}+1782109228317096yzw^{22}-2209033757z^{24}-878819780436z^{22}w^{2}-64503649553112z^{20}w^{4}-210563082143960z^{18}w^{6}+11070280904060973z^{16}w^{8}+84528119047158360z^{14}w^{10}+247579716071795072z^{12}w^{12}+364899304475652240z^{10}w^{14}+286883656461405297z^{8}w^{16}+118326578906455724z^{6}w^{18}+22275879657949848z^{4}w^{20}+691112983411368z^{2}w^{22}-205291994438489w^{24}}{w^{2}(5049220016yz^{21}-85108877560yz^{19}w^{2}-126698767272yz^{17}w^{4}+2519506167976yz^{15}w^{6}-945398859328yz^{13}w^{8}-13439284332768yz^{11}w^{10}+3431262261952yz^{9}w^{12}+27067130835584yz^{7}w^{14}-5140173282048yz^{5}w^{16}-16280904865792yz^{3}w^{18}+4338819824640yzw^{20}-6424325600z^{22}+24405436788z^{20}w^{2}+508046691880z^{18}w^{4}-1457966024977z^{16}w^{6}-4169312629944z^{14}w^{8}+7186977742840z^{12}w^{10}+15800317171520z^{10}w^{12}-11271721039152z^{8}w^{14}-20423866400000z^{6}w^{16}+3975037822336z^{4}w^{18}+7128061140480z^{2}w^{20}-650822973696w^{22})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.dx.2 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Equation of the image curve:

$0$ $=$ $ 4X^{4}+20X^{2}Y^{2}+49Y^{4}-2X^{2}Z^{2}-14Y^{2}Z^{2}+Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.96.0-16.y.2.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.bk.2.7 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.w.1.5 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.w.1.13 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-16.y.2.8 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-24.bk.2.7 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.bx.2.4 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.bx.2.7 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.1-48.bx.1.10 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1-48.bx.1.13 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1-48.by.2.5 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1-48.by.2.14 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1-48.ch.1.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1-48.ch.1.8 $48$ $2$ $2$ $1$ $0$ 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
48.576.17-48.ckq.1.2 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.768.17-48.blo.1.4 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$