Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K3 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}65&188\\22&141\end{bmatrix}$, $\begin{bmatrix}67&108\\186&157\end{bmatrix}$, $\begin{bmatrix}137&18\\22&157\end{bmatrix}$, $\begin{bmatrix}185&0\\0&161\end{bmatrix}$, $\begin{bmatrix}197&164\\100&111\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 204.192.3-204.j.1.1, 204.192.3-204.j.1.2, 204.192.3-204.j.1.3, 204.192.3-204.j.1.4, 204.192.3-204.j.1.5, 204.192.3-204.j.1.6, 204.192.3-204.j.1.7, 204.192.3-204.j.1.8, 204.192.3-204.j.1.9, 204.192.3-204.j.1.10, 204.192.3-204.j.1.11, 204.192.3-204.j.1.12, 204.192.3-204.j.1.13, 204.192.3-204.j.1.14, 204.192.3-204.j.1.15, 204.192.3-204.j.1.16 |
| Cyclic 204-isogeny field degree: | $36$ |
| Cyclic 204-torsion field degree: | $2304$ |
| Full 204-torsion field degree: | $3760128$ |
Rational points
This modular curve has no $\Q_p$ points for $p=23$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.1.d.1 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 204.24.0.g.1 | $204$ | $4$ | $4$ | $0$ | $?$ |
| 204.48.1.b.1 | $204$ | $2$ | $2$ | $1$ | $?$ |
| 204.48.1.c.1 | $204$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 204.192.5.o.1 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.o.2 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.o.3 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.o.4 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.y.1 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.y.2 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.y.3 | $204$ | $2$ | $2$ | $5$ |
| 204.192.5.y.4 | $204$ | $2$ | $2$ | $5$ |
| 204.288.13.bh.1 | $204$ | $3$ | $3$ | $13$ |