Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
Index: | $192$ | $\PSL_2$-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: | $1$ | ||||||
$\Q$-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: | 48.192.1.770 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&8\\40&17\end{bmatrix}$, $\begin{bmatrix}31&45\\36&7\end{bmatrix}$, $\begin{bmatrix}37&10\\12&35\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.96.1.ds.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $4$ |
Cyclic 48-torsion field degree: | $32$ |
Full 48-torsion field degree: | $6144$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + 5 y^{2} - z^{2} - w^{2} $ |
$=$ | $8 x^{2} - 2 y^{2} + z^{2}$ |
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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3}\cdot\frac{(81z^{8}+3240z^{6}w^{2}+4824z^{4}w^{4}+1440z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{2}(3z^{2}-2w^{2})^{8}(3z^{2}+2w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.96.0-16.x.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.96.0-24.bl.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-16.x.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.x.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.x.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-24.bl.2.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.bu.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.bu.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.1-48.bt.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bt.1.10 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bu.2.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bu.2.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.cf.1.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.cf.1.4 | $48$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
48.576.17-48.ckh.2.1 | $48$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
48.768.17-48.bld.2.3 | $48$ | $4$ | $4$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
96.384.5-96.de.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.dm.1.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.fq.2.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.fy.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |