Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}73&96\\186&1\end{bmatrix}$, $\begin{bmatrix}115&180\\174&181\end{bmatrix}$, $\begin{bmatrix}127&138\\144&151\end{bmatrix}$, $\begin{bmatrix}139&94\\114&107\end{bmatrix}$, $\begin{bmatrix}169&210\\180&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.144.5.a.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $40$ |
| Cyclic 228-torsion field degree: | $1440$ |
| Full 228-torsion field degree: | $1969920$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| 76.12.0.a.1 | $76$ | $24$ | $12$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(6)$ | $6$ | $2$ | $2$ | $1$ | $0$ |
| 228.96.1-228.a.1.11 | $228$ | $3$ | $3$ | $1$ | $?$ |
| 228.96.1-228.a.1.20 | $228$ | $3$ | $3$ | $1$ | $?$ |
| 228.144.1-6.a.1.3 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.144.1-228.m.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.144.1-228.m.1.5 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.144.3-228.mb.1.1 | $228$ | $2$ | $2$ | $3$ | $?$ |
| 228.144.3-228.mb.1.2 | $228$ | $2$ | $2$ | $3$ | $?$ |