Invariants
| Level: | $228$ | $\SL_2$-level: | $76$ | Newform level: | $1$ | ||
| Index: | $120$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot4\cdot19^{2}\cdot76$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 76A8 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}100&179\\41&48\end{bmatrix}$, $\begin{bmatrix}121&54\\0&175\end{bmatrix}$, $\begin{bmatrix}163&174\\10&175\end{bmatrix}$, $\begin{bmatrix}184&77\\25&46\end{bmatrix}$, $\begin{bmatrix}208&75\\3&52\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 228.240.8-228.c.1.1, 228.240.8-228.c.1.2, 228.240.8-228.c.1.3, 228.240.8-228.c.1.4, 228.240.8-228.c.1.5, 228.240.8-228.c.1.6, 228.240.8-228.c.1.7, 228.240.8-228.c.1.8, 228.240.8-228.c.1.9, 228.240.8-228.c.1.10, 228.240.8-228.c.1.11, 228.240.8-228.c.1.12, 228.240.8-228.c.1.13, 228.240.8-228.c.1.14, 228.240.8-228.c.1.15, 228.240.8-228.c.1.16 |
| Cyclic 228-isogeny field degree: | $8$ |
| Cyclic 228-torsion field degree: | $576$ |
| Full 228-torsion field degree: | $4727808$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| 12.6.0.c.1 | $12$ | $20$ | $20$ | $0$ | $0$ |
| $X_0(19)$ | $19$ | $6$ | $6$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.6.0.c.1 | $12$ | $20$ | $20$ | $0$ | $0$ |
| $X_0(38)$ | $38$ | $2$ | $2$ | $4$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.240.17.b.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.h.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.w.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.x.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.bf.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.bg.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.bh.1 | $228$ | $2$ | $2$ | $17$ |
| 228.240.17.bk.1 | $228$ | $2$ | $2$ | $17$ |
| 228.360.22.h.1 | $228$ | $3$ | $3$ | $22$ |
| 228.360.22.h.2 | $228$ | $3$ | $3$ | $22$ |
| 228.360.22.k.1 | $228$ | $3$ | $3$ | $22$ |