Invariants
| Level: | $208$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H0 |
Level structure
| $\GL_2(\Z/208\Z)$-generators: | $\begin{bmatrix}5&174\\2&89\end{bmatrix}$, $\begin{bmatrix}12&113\\65&180\end{bmatrix}$, $\begin{bmatrix}82&75\\171&90\end{bmatrix}$, $\begin{bmatrix}174&149\\99&8\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 208.48.0.bl.1 for the level structure with $-I$) |
| Cyclic 208-isogeny field degree: | $28$ |
| Cyclic 208-torsion field degree: | $1344$ |
| Full 208-torsion field degree: | $6709248$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-104.cb.2.8 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-16.e.1.3 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.o.1.5 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.o.1.6 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-104.cb.2.16 | $208$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 208.192.1-208.r.1.11 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.u.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bg.2.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bw.2.2 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.ci.2.1 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.cn.2.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.cz.2.2 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.dc.2.3 | $208$ | $2$ | $2$ | $1$ |