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Properties

Label 48.48.1.s.1
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $32$
Index: $48$ $\PSL_2$-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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.445

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&25\\40&45\end{bmatrix}$, $\begin{bmatrix}25&25\\24&35\end{bmatrix}$, $\begin{bmatrix}29&36\\20&35\end{bmatrix}$, $\begin{bmatrix}35&38\\8&43\end{bmatrix}$, $\begin{bmatrix}47&46\\24&47\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.s.1.1, 48.96.1-48.s.1.2, 48.96.1-48.s.1.3, 48.96.1-48.s.1.4, 48.96.1-48.s.1.5, 48.96.1-48.s.1.6, 48.96.1-48.s.1.7, 48.96.1-48.s.1.8, 48.96.1-48.s.1.9, 48.96.1-48.s.1.10, 48.96.1-48.s.1.11, 48.96.1-48.s.1.12, 48.96.1-48.s.1.13, 48.96.1-48.s.1.14, 48.96.1-48.s.1.15, 48.96.1-48.s.1.16, 96.96.1-48.s.1.1, 96.96.1-48.s.1.2, 96.96.1-48.s.1.3, 96.96.1-48.s.1.4, 96.96.1-48.s.1.5, 96.96.1-48.s.1.6, 96.96.1-48.s.1.7, 96.96.1-48.s.1.8, 240.96.1-48.s.1.1, 240.96.1-48.s.1.2, 240.96.1-48.s.1.3, 240.96.1-48.s.1.4, 240.96.1-48.s.1.5, 240.96.1-48.s.1.6, 240.96.1-48.s.1.7, 240.96.1-48.s.1.8, 240.96.1-48.s.1.9, 240.96.1-48.s.1.10, 240.96.1-48.s.1.11, 240.96.1-48.s.1.12, 240.96.1-48.s.1.13, 240.96.1-48.s.1.14, 240.96.1-48.s.1.15, 240.96.1-48.s.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

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 y + 6 x z + w^{2} $
$=$ $12 x^{2} - y^{2} - 3 y z - 3 z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 108 x^{4} - x^{2} y^{2} - x y 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{3}\cdot\frac{79272xz^{9}w^{2}+196704xz^{5}w^{6}-18432xzw^{10}-19656y^{2}z^{10}-27180y^{2}z^{6}w^{4}-17856y^{2}z^{2}w^{8}-58968yz^{11}-29196yz^{7}w^{4}-38784yz^{3}w^{8}-39285z^{12}+49572z^{8}w^{4}+6432z^{4}w^{8}-512w^{12}}{w^{4}(108xz^{5}w^{2}+60xzw^{6}+9y^{2}z^{6}+27y^{2}z^{2}w^{4}+27yz^{7}+57yz^{3}w^{4}+27z^{8}+69z^{4}w^{4}+4w^{8})}$

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
16.24.1.b.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.bh.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.g.1 $48$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.96.1.cn.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cn.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.co.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.co.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cp.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cp.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cq.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cq.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.co.1 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.zr.1 $48$ $4$ $4$ $9$ $0$ $1^{8}$
96.96.5.r.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.r.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bh.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bh.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bj.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bj.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cf.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cf.2 $96$ $2$ $2$ $5$ $?$ not computed
240.96.1.gr.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gr.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gs.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gs.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gt.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gt.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gu.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.gu.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.bi.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.cyi.1 $240$ $6$ $6$ $17$ $?$ not computed