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Properties

Label 48.96.1.bk.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: 48.96.1.1607

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&43\\24&47\end{bmatrix}$, $\begin{bmatrix}5&30\\32&25\end{bmatrix}$, $\begin{bmatrix}13&20\\8&41\end{bmatrix}$, $\begin{bmatrix}15&10\\32&11\end{bmatrix}$, $\begin{bmatrix}45&46\\8&21\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.bk.1.1, 48.192.1-48.bk.1.2, 48.192.1-48.bk.1.3, 48.192.1-48.bk.1.4, 48.192.1-48.bk.1.5, 48.192.1-48.bk.1.6, 48.192.1-48.bk.1.7, 48.192.1-48.bk.1.8, 48.192.1-48.bk.1.9, 48.192.1-48.bk.1.10, 48.192.1-48.bk.1.11, 48.192.1-48.bk.1.12, 96.192.1-48.bk.1.1, 96.192.1-48.bk.1.2, 96.192.1-48.bk.1.3, 96.192.1-48.bk.1.4, 96.192.1-48.bk.1.5, 96.192.1-48.bk.1.6, 96.192.1-48.bk.1.7, 96.192.1-48.bk.1.8, 240.192.1-48.bk.1.1, 240.192.1-48.bk.1.2, 240.192.1-48.bk.1.3, 240.192.1-48.bk.1.4, 240.192.1-48.bk.1.5, 240.192.1-48.bk.1.6, 240.192.1-48.bk.1.7, 240.192.1-48.bk.1.8, 240.192.1-48.bk.1.9, 240.192.1-48.bk.1.10, 240.192.1-48.bk.1.11, 240.192.1-48.bk.1.12
Cyclic 48-isogeny field degree: $4$
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 $ $=$ $ 3 x^{2} - 3 y^{2} - w^{2} $
$=$ $3 x^{2} + 3 y^{2} + 4 z^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{(16z^{8}-16z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{16}(2z-w)(2z+w)(4z^{2}+w^{2})}$

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.h.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.be.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.j.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bs.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bt.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bk.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bl.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.gw.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hb.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hd.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.he.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.288.17.ml.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.pb.2 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
96.192.5.bt.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bt.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bu.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bu.4 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.cy.3 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.cy.4 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.cz.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.cz.2 $96$ $2$ $2$ $9$ $?$ not computed
240.192.5.bsh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bsi.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bsl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bsm.1 $240$ $2$ $2$ $5$ $?$ not computed