Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D3 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}71&134\\156&175\end{bmatrix}$, $\begin{bmatrix}109&88\\90&185\end{bmatrix}$, $\begin{bmatrix}131&18\\156&77\end{bmatrix}$, $\begin{bmatrix}185&135\\120&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.72.3.jw.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $40$ |
| Cyclic 228-torsion field degree: | $2880$ |
| Full 228-torsion field degree: | $3939840$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.0-6.a.1.5 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 114.72.0-6.a.1.1 | $114$ | $2$ | $2$ | $0$ | $?$ |
| 228.48.1-228.n.1.7 | $228$ | $3$ | $3$ | $1$ | $?$ |
| 228.48.1-228.n.1.9 | $228$ | $3$ | $3$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.288.5-228.f.1.4 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.bc.1.4 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.cf.1.3 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.ci.1.6 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.eb.1.4 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.ec.1.3 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.ek.1.4 | $228$ | $2$ | $2$ | $5$ |
| 228.288.5-228.em.1.3 | $228$ | $2$ | $2$ | $5$ |