Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}5&192\\105&131\end{bmatrix}$, $\begin{bmatrix}91&12\\114&199\end{bmatrix}$, $\begin{bmatrix}115&2\\180&125\end{bmatrix}$, $\begin{bmatrix}163&134\\75&101\end{bmatrix}$, $\begin{bmatrix}181&30\\18&91\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 204.288.5-204.gz.1.1, 204.288.5-204.gz.1.2, 204.288.5-204.gz.1.3, 204.288.5-204.gz.1.4, 204.288.5-204.gz.1.5, 204.288.5-204.gz.1.6 |
| Cyclic 204-isogeny field degree: | $36$ |
| Cyclic 204-torsion field degree: | $2304$ |
| Full 204-torsion field degree: | $2506752$ |
Rational points
This modular curve has no $\Q_p$ points for $p=37$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.3.da.1 | $12$ | $2$ | $2$ | $3$ | $0$ |
| 102.72.1.c.1 | $102$ | $2$ | $2$ | $1$ | $?$ |
| 204.48.1.bu.1 | $204$ | $3$ | $3$ | $1$ | $?$ |
| 204.72.1.p.1 | $204$ | $2$ | $2$ | $1$ | $?$ |
| 204.72.1.cp.1 | $204$ | $2$ | $2$ | $1$ | $?$ |
| 204.72.3.lt.1 | $204$ | $2$ | $2$ | $3$ | $?$ |
| 204.72.3.lx.1 | $204$ | $2$ | $2$ | $3$ | $?$ |
| 204.72.3.rk.1 | $204$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 204.288.13.es.1 | $204$ | $2$ | $2$ | $13$ |
| 204.288.13.et.1 | $204$ | $2$ | $2$ | $13$ |
| 204.288.13.eu.1 | $204$ | $2$ | $2$ | $13$ |
| 204.288.13.ev.1 | $204$ | $2$ | $2$ | $13$ |