Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 12E0 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}11&76\\78&97\end{bmatrix}$, $\begin{bmatrix}107&208\\98&117\end{bmatrix}$, $\begin{bmatrix}119&216\\24&41\end{bmatrix}$, $\begin{bmatrix}134&145\\131&144\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.24.0.r.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $40$ |
| Cyclic 228-torsion field degree: | $1440$ |
| Full 228-torsion field degree: | $11819520$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 114.24.0-6.a.1.4 | $114$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.96.1-228.d.1.19 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.g.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.y.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bb.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bg.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bj.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bs.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bv.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.144.1-228.q.1.2 | $228$ | $3$ | $3$ | $1$ |