Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{3}\cdot12^{3}$ | Cusp orbits | $3^{2}$ | ||
| 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: | 12F2 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}45&47\\55&224\end{bmatrix}$, $\begin{bmatrix}102&197\\193&5\end{bmatrix}$, $\begin{bmatrix}117&112\\82&87\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.48.2.f.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $120$ |
| Cyclic 228-torsion field degree: | $8640$ |
| Full 228-torsion field degree: | $5909760$ |
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
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 | $3$ | $12$ | $12$ | $0$ | $0$ |
| 76.12.0-38.a.1.4 | $76$ | $8$ | $8$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.32.0-12.a.1.1 | $12$ | $3$ | $3$ | $0$ | $0$ |
| 114.48.0-114.a.1.3 | $114$ | $2$ | $2$ | $0$ | $?$ |
| 228.48.0-114.a.1.8 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.288.7-228.fq.1.8 | $228$ | $3$ | $3$ | $7$ |
| 228.384.5-228.bi.2.8 | $228$ | $4$ | $4$ | $5$ |