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Properties

Label 48.96.1.di.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: 48.96.1.1839

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&26\\32&29\end{bmatrix}$, $\begin{bmatrix}13&25\\4&9\end{bmatrix}$, $\begin{bmatrix}15&22\\4&29\end{bmatrix}$, $\begin{bmatrix}23&2\\0&43\end{bmatrix}$, $\begin{bmatrix}29&23\\24&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.di.1.1, 48.192.1-48.di.1.2, 48.192.1-48.di.1.3, 48.192.1-48.di.1.4, 48.192.1-48.di.1.5, 48.192.1-48.di.1.6, 48.192.1-48.di.1.7, 48.192.1-48.di.1.8, 48.192.1-48.di.1.9, 48.192.1-48.di.1.10, 48.192.1-48.di.1.11, 48.192.1-48.di.1.12, 96.192.1-48.di.1.1, 96.192.1-48.di.1.2, 96.192.1-48.di.1.3, 96.192.1-48.di.1.4, 96.192.1-48.di.1.5, 96.192.1-48.di.1.6, 96.192.1-48.di.1.7, 96.192.1-48.di.1.8, 240.192.1-48.di.1.1, 240.192.1-48.di.1.2, 240.192.1-48.di.1.3, 240.192.1-48.di.1.4, 240.192.1-48.di.1.5, 240.192.1-48.di.1.6, 240.192.1-48.di.1.7, 240.192.1-48.di.1.8, 240.192.1-48.di.1.9, 240.192.1-48.di.1.10, 240.192.1-48.di.1.11, 240.192.1-48.di.1.12
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
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 $ $=$ $ 2 x^{2} + 2 x y + 2 y^{2} - z^{2} $
$=$ $2 x^{2} + 2 x y - 4 y^{2} - 5 z^{2} + 3 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 441 x^{4} - 60 x^{2} y^{2} - 42 x^{2} z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 3x$
$\displaystyle Z$ $=$ $\displaystyle 3w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3}\cdot\frac{(16z^{8}-384z^{6}w^{2}+720z^{4}w^{4}-432z^{2}w^{6}+81w^{8})^{3}}{w^{2}z^{16}(4z^{2}-3w^{2})^{2}(8z^{2}-3w^{2})}$

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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.v.2 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bp.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.ba.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bl.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bp.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.x.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bn.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
48.192.5.jg.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.jh.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ji.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.jj.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.bbz.2 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.xz.1 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$
96.192.5.ej.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ek.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.er.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.es.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.ha.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.hi.1 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.kc.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.kk.1 $96$ $2$ $2$ $9$ $?$ not computed
240.192.5.cdk.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cdl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cdm.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cdn.1 $240$ $2$ $2$ $5$ $?$ not computed