Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
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: | 16E1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1498 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}25&3\\32&43\end{bmatrix}$, $\begin{bmatrix}37&4\\0&13\end{bmatrix}$, $\begin{bmatrix}37&14\\12&35\end{bmatrix}$, $\begin{bmatrix}45&29\\20&33\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.1.w.1 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: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x z - w^{2} $ |
$=$ | $48 x^{2} + y^{2} - 3 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 3 x^{2} y^{2} - z^{4} $ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{y^{12}-672y^{8}w^{4}+162048y^{4}w^{8}+2985255z^{12}-15863040z^{8}w^{4}+28200960z^{4}w^{8}-16769024w^{12}}{w^{4}(y^{8}+48y^{4}w^{4}-81z^{8}+144z^{4}w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.w.1 :
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
$0$ | $=$ | $ 9X^{4}-3X^{2}Y^{2}-Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0-24.bl.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.g.1.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-48.g.1.26 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0-24.bl.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1-16.b.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.dd.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.dd.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.de.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.de.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.df.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.df.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.dg.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.192.1-48.dg.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.288.9-48.cs.1.21 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
48.384.9-48.zv.1.5 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
96.192.5-96.t.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.t.2.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bl.1.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bl.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bn.1.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.bn.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.ch.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5-96.ch.2.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.192.1-240.hh.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hh.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hi.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hi.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hj.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hj.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hk.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.192.1-240.hk.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.480.17-240.bm.1.17 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |