Properties

Label 48.96.1-48.cc.2.5
Level $48$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $96$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{4}$
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: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.922

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}17&28\\4&19\end{bmatrix}$, $\begin{bmatrix}23&17\\8&9\end{bmatrix}$, $\begin{bmatrix}31&38\\44&45\end{bmatrix}$, $\begin{bmatrix}41&15\\44&29\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.48.1.cc.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
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 $ $=$ $ 8 x^{2} - 2 x z + y^{2} $
$=$ $8 x^{2} + 22 x z + y^{2} + 6 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} + x^{2} y^{2} - 9 x^{2} z^{2} + 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 to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^5\,\frac{1492992y^{12}-746496y^{10}w^{2}-404352y^{8}w^{4}+76032y^{6}w^{6}+54864y^{4}w^{8}+9576y^{2}w^{10}-1469664z^{12}+1819584z^{10}w^{2}+1318032z^{8}w^{4}+1963440z^{6}w^{6}-368874z^{4}w^{8}+7560z^{2}w^{10}+725w^{12}}{w^{2}(20736y^{8}w^{2}-3456y^{6}w^{4}-24y^{2}w^{8}-15552z^{10}+1296z^{8}w^{2}+3456z^{6}w^{4}-1296z^{4}w^{6}+168z^{2}w^{8}-7w^{10})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.cc.2 :

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 3y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Equation of the image curve:

$0$ $=$ $ 9X^{4}+X^{2}Y^{2}-9X^{2}Z^{2}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.48.0-16.f.2.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0-24.by.1.10 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0-16.f.2.8 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0-24.by.1.5 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1-48.b.1.3 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1-48.b.1.12 $48$ $2$ $2$ $1$ $0$ 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
48.192.1-48.p.1.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.bb.1.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.bq.2.3 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.bz.2.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.dn.1.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.ea.1.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.ee.2.3 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.ep.1.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.288.9-48.ju.2.9 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.384.9-48.bfv.2.7 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.192.1-240.or.2.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.oz.1.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.px.2.5 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.qf.1.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.tp.2.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.tx.1.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.uv.2.5 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.vd.1.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.480.17-240.fo.1.6 $240$ $5$ $5$ $17$ $?$ not computed