Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 | $2^{2}\cdot4^{3}$ | ||
| 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: | 48.192.1.1636 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&40\\0&37\end{bmatrix}$, $\begin{bmatrix}7&8\\36&17\end{bmatrix}$, $\begin{bmatrix}13&20\\32&9\end{bmatrix}$, $\begin{bmatrix}43&34\\28&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.a.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - 2 z^{2} - w^{2} $ |
| $=$ | $3 x y + 3 y^{2} + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} + 9 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z^{8}-4z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{16}(2z^{2}-w^{2})(2z^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-3X^{2}Y^{2}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.a.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.ba.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.d.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.d.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.ba.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bw.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bw.2.12 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.by.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.by.1.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-16.a.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bg.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bg.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bi.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bi.2.12 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.384.5-48.y.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.ba.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.bb.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.bc.1.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.576.17-48.dg.1.20 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.hn.2.10 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 240.384.5-240.bdd.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bde.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bdf.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bdg.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |