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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
Level structure
| $\GL_2(\Z/208\Z)$-generators: | $\begin{bmatrix}27&136\\144&49\end{bmatrix}$, $\begin{bmatrix}65&32\\132&195\end{bmatrix}$, $\begin{bmatrix}73&64\\70&185\end{bmatrix}$, $\begin{bmatrix}87&136\\16&179\end{bmatrix}$, $\begin{bmatrix}103&88\\150&149\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 208.48.0.f.2 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 |
|---|---|---|---|---|---|
| 16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-8.i.1.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.m.2.2 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.m.2.31 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.n.1.2 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-208.n.1.31 | $208$ | $2$ | $2$ | $0$ | $?$ |
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.a.2.4 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.b.2.13 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.c.1.10 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.d.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.e.1.13 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.f.2.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.g.2.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.h.2.14 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.i.1.9 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.j.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.k.2.13 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.l.2.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.3-208.cc.2.16 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-208.ce.2.14 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-208.cj.2.12 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-208.cm.2.10 | $208$ | $2$ | $2$ | $3$ |