Invariants
| Level: | $208$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8I0 |
Level structure
| $\GL_2(\Z/208\Z)$-generators: | $\begin{bmatrix}52&185\\193&28\end{bmatrix}$, $\begin{bmatrix}128&49\\199&122\end{bmatrix}$, $\begin{bmatrix}134&169\\83&68\end{bmatrix}$, $\begin{bmatrix}183&100\\194&201\end{bmatrix}$, $\begin{bmatrix}204&75\\77&82\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.1 for the level structure with $-I$) |
| Cyclic 208-isogeny field degree: | $28$ |
| Cyclic 208-torsion field degree: | $2688$ |
| Full 208-torsion field degree: | $13418496$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 138 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{x^{24}(x^{8}+4x^{6}y^{2}-10x^{4}y^{4}-28x^{2}y^{6}+y^{8})^{3}}{y^{4}x^{26}(x^{2}+y^{2})^{8}(x^{2}+2y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 208.24.0-8.n.1.1 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.24.0-8.n.1.3 | $208$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 208.96.0-8.j.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-8.m.2.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-8.n.2.6 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-8.o.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-16.u.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-16.w.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-16.y.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-16.ba.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.bi.1.8 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.bk.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.bm.1.6 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bm.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.bo.2.6 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bo.1.7 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bu.2.13 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bw.2.13 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-16.q.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-16.s.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-16.u.1.2 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-16.w.1.2 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.bs.2.13 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.bu.2.13 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.ca.1.7 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.cc.1.7 | $208$ | $2$ | $2$ | $1$ |