Invariants
| Level: | $208$ | $\SL_2$-level: | $16$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/208\Z)$-generators: | $\begin{bmatrix}0&147\\63&36\end{bmatrix}$, $\begin{bmatrix}94&57\\139&12\end{bmatrix}$, $\begin{bmatrix}117&84\\50&63\end{bmatrix}$, $\begin{bmatrix}129&158\\18&197\end{bmatrix}$, $\begin{bmatrix}183&102\\0&21\end{bmatrix}$, $\begin{bmatrix}188&135\\91&128\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 208-isogeny field degree: | $28$ |
| Cyclic 208-torsion field degree: | $2688$ |
| Full 208-torsion field degree: | $26836992$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 208.48.0-16.e.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.e.1.11 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.e.2.4 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.e.2.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.1.10 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.2.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.f.2.9 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.g.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.g.1.9 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.h.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-16.h.1.15 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.i.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.k.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.1.9 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.1.32 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.2.11 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.m.2.30 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.1.11 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.1.30 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.2.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.n.2.32 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.o.1.15 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.o.1.22 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.p.1.10 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.p.1.31 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.q.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.r.1.4 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.ba.1.6 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.ba.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.ba.2.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.ba.2.8 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.bb.1.4 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.bb.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.bb.2.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-8.bb.2.8 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.bj.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.bl.1.4 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.bn.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.bp.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.ca.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.ca.1.11 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.ca.2.7 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.ca.2.12 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.cb.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.cb.1.10 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.cb.2.6 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-104.cb.2.15 | $208$ | $2$ | $2$ | $0$ |
| 208.48.1-16.a.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-16.a.1.15 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.a.1.10 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.a.1.31 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-16.b.1.7 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-16.b.1.9 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.b.1.15 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.b.1.22 | $208$ | $2$ | $2$ | $1$ |
| 208.336.11-104.bx.1.2 | $208$ | $14$ | $14$ | $11$ |