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 | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | 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: | 12J0 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}43&62\\204&209\end{bmatrix}$, $\begin{bmatrix}115&78\\48&97\end{bmatrix}$, $\begin{bmatrix}142&21\\123&76\end{bmatrix}$, $\begin{bmatrix}180&133\\103&174\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.48.0.c.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $20$ |
| Cyclic 228-torsion field degree: | $1440$ |
| 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 |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ |
| 76.12.0-4.c.1.2 | $76$ | $8$ | $8$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.0-12.g.1.10 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 228.48.0-12.g.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 |
|---|---|---|---|---|
| 228.192.1-228.b.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.i.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.j.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.k.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.l.4.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.m.2.2 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.n.2.4 | $228$ | $2$ | $2$ | $1$ |
| 228.192.1-228.o.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.288.3-228.c.1.2 | $228$ | $3$ | $3$ | $3$ |