Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.0.92 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&1\\8&15\end{bmatrix}$, $\begin{bmatrix}9&14\\0&11\end{bmatrix}$, $\begin{bmatrix}13&15\\0&11\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.0.f.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $256$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 4 x^{2} + y^{2} + z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.k.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.f.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.f.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.f.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.f.2.13 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-8.k.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.