Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E5 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}117&118\\10&201\end{bmatrix}$, $\begin{bmatrix}159&158\\200&9\end{bmatrix}$, $\begin{bmatrix}223&122\\10&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.192.5.r.4 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $20$ |
| Cyclic 228-torsion field degree: | $1440$ |
| Full 228-torsion field degree: | $1477440$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.192.3-12.g.2.5 | $12$ | $2$ | $2$ | $3$ | $0$ |
| 228.192.1-228.b.2.6 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.1-228.b.2.10 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.1-228.e.4.2 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.1-228.e.4.14 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.1-228.f.2.6 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.1-228.f.2.16 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.192.3-228.c.1.10 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-228.c.1.13 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-12.g.2.4 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-228.s.2.11 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-228.s.2.14 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-228.t.2.11 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.192.3-228.t.2.12 | $228$ | $2$ | $2$ | $3$ | $?$ |