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Properties

Label 48.96.1.ek.1
Level $48$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
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: $1$
$\Q$-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: 48.96.1.880

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}13&31\\8&3\end{bmatrix}$, $\begin{bmatrix}23&24\\4&1\end{bmatrix}$, $\begin{bmatrix}41&9\\12&13\end{bmatrix}$, $\begin{bmatrix}45&35\\20&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.ek.1.1, 48.192.1-48.ek.1.2, 48.192.1-48.ek.1.3, 48.192.1-48.ek.1.4, 48.192.1-48.ek.1.5, 48.192.1-48.ek.1.6, 48.192.1-48.ek.1.7, 48.192.1-48.ek.1.8, 240.192.1-48.ek.1.1, 240.192.1-48.ek.1.2, 240.192.1-48.ek.1.3, 240.192.1-48.ek.1.4, 240.192.1-48.ek.1.5, 240.192.1-48.ek.1.6, 240.192.1-48.ek.1.7, 240.192.1-48.ek.1.8
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

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

$ 0 $ $=$ $ x^{2} - 2 y z $
$=$ $7 x^{2} - 3 y^{2} + 4 y z - 3 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 12 x^{2} z^{2} - 3 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}w$
$\displaystyle Z$ $=$ $\displaystyle 2z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^4}\cdot\frac{591366152471592960yz^{23}+871123745469579264yz^{21}w^{2}+475879741758185472yz^{19}w^{4}+117960369711943680yz^{17}w^{6}+13577635223635968yz^{15}w^{8}+842110162311168yz^{13}w^{10}+30382919451648yz^{11}w^{12}+651602299392yz^{9}w^{14}+8112965184yz^{7}w^{16}+54264384yz^{5}w^{18}+163872yz^{3}w^{20}+144yzw^{22}-101462391107334144z^{24}-114614620997345280z^{22}w^{2}-30679581853089792z^{20}w^{4}+7275542399963136z^{18}w^{6}+4340171459365632z^{16}w^{8}+588982591669248z^{14}w^{10}+37433117316096z^{12}w^{12}+1300347136512z^{10}w^{14}+25611644016z^{8}w^{16}+277401888z^{6}w^{18}+1484352z^{4}w^{20}+3024z^{2}w^{22}+w^{24}}{w^{2}z^{8}(13048147004544yz^{13}+1645649662176yz^{11}w^{2}+79272155952yz^{9}w^{4}+1816682904yz^{7}w^{6}+20000088yz^{5}w^{8}+92808yz^{3}w^{10}+120yzw^{12}-2238708098304z^{14}+486520591536z^{12}w^{2}+75360827952z^{10}w^{4}+3516205545z^{8}w^{6}+71658540z^{6}w^{8}+639918z^{4}w^{10}+2028z^{2}w^{12}+w^{14})}$

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
16.48.0.z.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bn.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.y.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bw.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bo.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.br.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.ck.1 $48$ $2$ $2$ $1$ $1$ 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.clp.2 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.384.17.bmm.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$