Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-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
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}7&172\\216&179\end{bmatrix}$, $\begin{bmatrix}99&40\\92&119\end{bmatrix}$, $\begin{bmatrix}165&136\\122&191\end{bmatrix}$, $\begin{bmatrix}193&78\\198&127\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.96.3.a.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $40$ |
| Cyclic 228-torsion field degree: | $2880$ |
| Full 228-torsion field degree: | $2954880$ |
Rational points
This modular curve has no $\Q_p$ points for $p=23$, and therefore no 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 |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
| 76.24.0.a.1 | $76$ | $8$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.1-12.a.1.12 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 228.96.1-12.a.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.96.1-228.a.1.8 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.96.1-228.a.1.11 | $228$ | $2$ | $2$ | $1$ | $?$ |
| 228.96.2-228.g.1.4 | $228$ | $2$ | $2$ | $2$ | $?$ |
| 228.96.2-228.g.1.13 | $228$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.384.5-228.f.1.4 | $228$ | $2$ | $2$ | $5$ |
| 228.384.5-228.f.2.2 | $228$ | $2$ | $2$ | $5$ |
| 228.384.5-228.p.1.3 | $228$ | $2$ | $2$ | $5$ |
| 228.384.5-228.p.2.4 | $228$ | $2$ | $2$ | $5$ |