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: | $\begin{bmatrix}18&53\\173&90\end{bmatrix}$, $\begin{bmatrix}140&207\\221&52\end{bmatrix}$, $\begin{bmatrix}191&130\\206&57\end{bmatrix}$, $\begin{bmatrix}209&160\\6&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.i.1 for the level structure with $-I$) |
| Cyclic 228-isogeny field degree: | $40$ |
| Cyclic 228-torsion field degree: | $2880$ |
| 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, including 85 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^6}\cdot\frac{(3x-2y)^{3}(3x+y)^{24}(3x+2y)^{3}(9x^{3}-18x^{2}y+12xy^{2}+8y^{3})^{3}(9x^{3}+18x^{2}y+12xy^{2}-8y^{3})^{3}}{y^{4}x^{12}(3x+y)^{24}(3x^{2}-4y^{2})^{3}(27x^{2}-4y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 228.24.0-6.a.1.1 | $228$ | $2$ | $2$ | $0$ | $?$ |
| 228.24.0-6.a.1.7 | $228$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.96.1-12.c.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.f.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.n.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.o.1.3 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.u.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.v.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bc.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bd.1.3 | $228$ | $2$ | $2$ | $1$ |
| 228.144.1-12.i.1.2 | $228$ | $3$ | $3$ | $1$ |