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Properties

Label 48.96.1.dt.2
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 $2^{2}\cdot4^{3}$
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: 48.96.1.998

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&5\\8&5\end{bmatrix}$, $\begin{bmatrix}9&29\\32&31\end{bmatrix}$, $\begin{bmatrix}25&43\\4&21\end{bmatrix}$, $\begin{bmatrix}27&10\\44&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.dt.2.1, 48.192.1-48.dt.2.2, 48.192.1-48.dt.2.3, 48.192.1-48.dt.2.4, 48.192.1-48.dt.2.5, 48.192.1-48.dt.2.6, 48.192.1-48.dt.2.7, 48.192.1-48.dt.2.8, 240.192.1-48.dt.2.1, 240.192.1-48.dt.2.2, 240.192.1-48.dt.2.3, 240.192.1-48.dt.2.4, 240.192.1-48.dt.2.5, 240.192.1-48.dt.2.6, 240.192.1-48.dt.2.7, 240.192.1-48.dt.2.8
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 $ $=$ $ 4 x^{2} - 2 y z + w^{2} $
$=$ $2 x^{2} - y^{2} + 3 y z - z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 8 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} + 2 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^{11}\,\frac{1079070720yz^{23}-2967444480yz^{21}w^{2}+3322805760yz^{19}w^{4}-1912354560yz^{17}w^{6}+573298176yz^{15}w^{8}-70448448yz^{13}w^{10}-3533088yz^{11}w^{12}+1393200yz^{9}w^{14}-10848yz^{7}w^{16}-8688yz^{5}w^{18}-24yz^{3}w^{20}+12yzw^{22}-289136128z^{24}+327873024z^{22}w^{2}+336191616z^{20}w^{4}-780387200z^{18}w^{6}+530198304z^{16}w^{8}-159871392z^{14}w^{10}+16040440z^{12}w^{12}+1792536z^{10}w^{14}-342540z^{8}w^{16}-11272z^{6}w^{18}+1758z^{4}w^{20}+54z^{2}w^{22}-w^{24}}{w^{4}(619790336yz^{19}-1394528256yz^{17}w^{2}+1321226240yz^{15}w^{4}-685196288yz^{13}w^{6}+211640832yz^{11}w^{8}-39672064yz^{9}w^{10}+4398976yz^{7}w^{12}-267264yz^{5}w^{14}+7520yz^{3}w^{16}-64yzw^{18}-166072320z^{20}+105285632z^{18}w^{2}+216279808z^{16}w^{4}-321415680z^{14}w^{6}+184725632z^{12}w^{8}-56824448z^{10}w^{10}+9949200z^{8}w^{12}-968896z^{6}w^{14}+47464z^{4}w^{16}-912z^{2}w^{18}+3w^{20})}$

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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.y.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bi.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.u.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.by.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bo.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bs.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.cg.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.cki.1 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.384.17.blf.1 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$