Invariants
| Level: | $228$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}15&73\\212&37\end{bmatrix}$, $\begin{bmatrix}57&85\\223&84\end{bmatrix}$, $\begin{bmatrix}169&218\\171&125\end{bmatrix}$, $\begin{bmatrix}193&110\\145&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.8.0.a.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $120$ |
| Cyclic 228-torsion field degree: | $8640$ |
| Full 228-torsion field degree: | $35458560$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 222 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6}\cdot\frac{x^{8}(x^{2}-108y^{2})(x^{2}-12y^{2})^{3}}{y^{6}x^{10}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.2.0.a.1 | $4$ | $8$ | $4$ | $0$ | $0$ |
| 57.8.0-3.a.1.2 | $57$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 57.8.0-3.a.1.2 | $57$ | $2$ | $2$ | $0$ | $0$ |
| 228.8.0-3.a.1.4 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.32.0-12.b.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.32.0-12.b.2.2 | $228$ | $2$ | $2$ | $0$ |
| 228.32.0-228.b.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.32.0-228.b.2.6 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.d.1.3 | $228$ | $3$ | $3$ | $0$ |
| 228.48.1-12.b.1.1 | $228$ | $3$ | $3$ | $1$ |
| 228.64.1-12.b.1.2 | $228$ | $4$ | $4$ | $1$ |
| 228.320.11-228.a.1.24 | $228$ | $20$ | $20$ | $11$ |