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Properties

Label 48.96.1.dl.2
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: $288$
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: 48.96.1.1048

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&21\\8&19\end{bmatrix}$, $\begin{bmatrix}13&36\\12&7\end{bmatrix}$, $\begin{bmatrix}29&41\\16&23\end{bmatrix}$, $\begin{bmatrix}43&18\\0&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.dl.2.1, 48.192.1-48.dl.2.2, 48.192.1-48.dl.2.3, 48.192.1-48.dl.2.4, 48.192.1-48.dl.2.5, 48.192.1-48.dl.2.6, 48.192.1-48.dl.2.7, 48.192.1-48.dl.2.8, 96.192.1-48.dl.2.1, 96.192.1-48.dl.2.2, 96.192.1-48.dl.2.3, 96.192.1-48.dl.2.4, 240.192.1-48.dl.2.1, 240.192.1-48.dl.2.2, 240.192.1-48.dl.2.3, 240.192.1-48.dl.2.4, 240.192.1-48.dl.2.5, 240.192.1-48.dl.2.6, 240.192.1-48.dl.2.7, 240.192.1-48.dl.2.8
Cyclic 48-isogeny field degree: $4$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

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

$ 0 $ $=$ $ 9 x^{4} - 24 x^{2} y^{2} + 3 x^{2} z^{2} + 4 y^{4} - 4 y^{2} z^{2} + 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{3}\cdot\frac{1120043793960xz^{23}+2053413622260xz^{21}w^{2}+1532877791580xz^{19}w^{4}+588138668820xz^{17}w^{6}+117544041648xz^{15}w^{8}+9629422236xz^{13}w^{10}-321952644xz^{11}w^{12}-84636900xz^{9}w^{14}-439344xz^{7}w^{16}+234576xz^{5}w^{18}-432xz^{3}w^{20}-144xzw^{22}-300114830079z^{24}-226881728838z^{22}w^{2}+155092021353z^{20}w^{4}+240005644650z^{18}w^{6}+108707221017z^{16}w^{8}+21852420894z^{14}w^{10}+1461685095z^{12}w^{12}-108896562z^{10}w^{14}-13872870z^{8}w^{16}+304344z^{6}w^{18}+31644z^{4}w^{20}-648z^{2}w^{22}-8w^{24}}{w^{4}(11913411312xz^{19}+17870116968xz^{17}w^{2}+11287194480xz^{15}w^{4}+3902406984xz^{13}w^{6}+803573784xz^{11}w^{8}+100419912xz^{9}w^{10}+7423272xz^{7}w^{12}+300672xz^{5}w^{14}+5640xz^{3}w^{16}+32xzw^{18}-3192188940z^{20}-1349177796z^{18}w^{2}+1847671641z^{16}w^{4}+1830562740z^{14}w^{6}+701380134z^{12}w^{8}+143836884z^{10}w^{10}+16789275z^{8}w^{12}+1090008z^{6}w^{14}+35598z^{4}w^{16}+456z^{2}w^{18}+w^{20})}$

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
16.48.0.u.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bi.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.u.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bu.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bw.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ca.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ce.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
48.288.17.cjs.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.bkp.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.cx.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.df.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.fj.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.fr.2 $96$ $2$ $2$ $5$ $?$ not computed