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 | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.1.2117 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&47\\16&19\end{bmatrix}$, $\begin{bmatrix}7&35\\12&31\end{bmatrix}$, $\begin{bmatrix}37&47\\32&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.cl.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| 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 $ | $=$ | $ x^{2} + x y + y^{2} + z^{2} $ |
| $=$ | $6 y^{2} - 8 z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 441 x^{4} + 30 x^{2} y^{2} - 84 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^8\cdot3}\cdot\frac{(256z^{8}-3072z^{6}w^{2}+2880z^{4}w^{4}-864z^{2}w^{6}+81w^{8})^{3}}{w^{2}z^{16}(8z^{2}-3w^{2})^{2}(16z^{2}-3w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.cl.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 441X^{4}+30X^{2}Y^{2}+Y^{4}-84X^{2}Z^{2}-2Y^{2}Z^{2}+4Z^{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.s.2.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.s.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.bk.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.bk.2.6 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.x.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.x.1.10 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bg.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bg.2.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bj.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bj.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-48.r.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.r.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bj.2.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bj.2.14 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.576.17-48.bai.1.2 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.wi.1.4 | $48$ | $4$ | $4$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |