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 $4$ are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| 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: | 16G0 |
Level structure
| $\GL_2(\Z/208\Z)$-generators: | $\begin{bmatrix}1&16\\163&39\end{bmatrix}$, $\begin{bmatrix}55&0\\2&139\end{bmatrix}$, $\begin{bmatrix}81&16\\137&43\end{bmatrix}$, $\begin{bmatrix}113&176\\73&111\end{bmatrix}$, $\begin{bmatrix}155&72\\35&173\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.48.0.j.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, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(x-y)^{48}(x^{8}-24x^{7}y+144x^{6}y^{2}-304x^{5}y^{3}+136x^{4}y^{4}+288x^{3}y^{5}-320x^{2}y^{6}+64xy^{7}+16y^{8})^{3}(x^{8}+8x^{7}y-80x^{6}y^{2}+144x^{5}y^{3}+136x^{4}y^{4}-608x^{3}y^{5}+576x^{2}y^{6}-192xy^{7}+16y^{8})^{3}}{y^{2}x^{2}(x-2y)^{2}(x-y)^{50}(x^{2}-2y^{2})^{2}(x^{2}-4xy+2y^{2})^{2}(x^{2}-2xy+2y^{2})^{16}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 104.48.0-8.q.1.3 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-16.e.2.2 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-16.e.2.7 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-16.e.2.10 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-16.e.2.15 | $208$ | $2$ | $2$ | $0$ | $?$ |
| 208.48.0-8.q.1.4 | $208$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 208.192.1-16.l.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-16.l.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-16.l.2.5 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-16.m.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-16.m.1.10 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-16.m.2.10 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bf.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bf.1.8 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bf.2.10 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bg.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bg.1.8 | $208$ | $2$ | $2$ | $1$ |
| 208.192.1-208.bg.2.11 | $208$ | $2$ | $2$ | $1$ |
| 208.192.3-16.ck.1.4 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-16.cl.1.4 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-208.gk.1.4 | $208$ | $2$ | $2$ | $3$ |
| 208.192.3-208.gl.1.8 | $208$ | $2$ | $2$ | $3$ |