Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}15&70\\88&109\end{bmatrix}$, $\begin{bmatrix}39&200\\34&45\end{bmatrix}$, $\begin{bmatrix}61&49\\48&107\end{bmatrix}$, $\begin{bmatrix}73&97\\174&191\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 228-isogeny field degree: | $160$ |
| Cyclic 228-torsion field degree: | $11520$ |
| Full 228-torsion field degree: | $15759360$ |
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 |
|---|---|---|---|---|---|
| 12.18.1.e.1 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 114.18.1.b.1 | $114$ | $2$ | $2$ | $1$ | $?$ |
| 228.12.0.m.1 | $228$ | $3$ | $3$ | $0$ | $?$ |
| 228.18.0.j.1 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.72.3.hs.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.hu.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.ia.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.ic.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.ku.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.kw.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.lc.1 | $228$ | $2$ | $2$ | $3$ |
| 228.72.3.le.1 | $228$ | $2$ | $2$ | $3$ |