Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12I0 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}59&66\\44&73\end{bmatrix}$, $\begin{bmatrix}103&26\\102&197\end{bmatrix}$, $\begin{bmatrix}155&40\\92&45\end{bmatrix}$, $\begin{bmatrix}187&30\\108&19\end{bmatrix}$, $\begin{bmatrix}227&36\\182&181\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.48.0.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: | $5909760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 76.12.0-2.a.1.1 | $76$ | $8$ | $8$ | $0$ | $?$ |
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 228.48.0-6.a.1.2 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.192.1-228.a.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.b.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.b.4.8 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.c.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.c.2.8 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.d.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.d.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.e.2.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.e.4.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.f.2.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.f.3.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.g.3.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.g.4.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.h.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.h.4.8 | $228$ | $2$ | $2$ | $1$ |
| 228.192.3-228.k.1.4 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.l.1.4 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.m.2.8 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.n.1.12 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.s.2.8 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.t.2.8 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.u.1.6 | $228$ | $2$ | $2$ | $3$ |
| 228.192.3-228.v.1.4 | $228$ | $2$ | $2$ | $3$ |
| 228.288.3-228.a.1.13 | $228$ | $3$ | $3$ | $3$ |