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: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6I0 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}69&92\\128&195\end{bmatrix}$, $\begin{bmatrix}91&126\\10&101\end{bmatrix}$, $\begin{bmatrix}131&138\\192&155\end{bmatrix}$, $\begin{bmatrix}147&92\\152&87\end{bmatrix}$, $\begin{bmatrix}169&188\\156&29\end{bmatrix}$, $\begin{bmatrix}221&220\\114&151\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.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: | $11819520$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 76.12.0-2.a.1.1 | $76$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 76.12.0-2.a.1.1 | $76$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 228.96.0-12.a.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.1.11 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.1.13 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.1.14 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.1 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.2 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.5 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.11 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.12 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-12.a.2.15 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.9 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.11 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.20 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.29 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.1.31 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.4 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.5 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.7 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.28 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.29 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.31 | $228$ | $2$ | $2$ | $0$ |
| 228.96.1-12.a.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.a.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.a.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.a.1.6 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.a.1.18 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.b.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.b.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.b.1.11 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.b.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.b.1.12 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.b.1.13 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.c.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.c.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.c.1.8 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.c.1.6 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.c.1.12 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.c.1.18 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.d.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.d.1.5 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-12.d.1.6 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.d.1.6 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.d.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.d.1.18 | $228$ | $2$ | $2$ | $1$ |
| 228.96.2-12.a.1.1 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-12.a.1.2 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-12.a.2.1 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-12.a.2.5 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.a.1.7 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.a.1.9 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.a.2.3 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.a.2.5 | $228$ | $2$ | $2$ | $2$ |
| 228.144.1-6.a.1.2 | $228$ | $3$ | $3$ | $1$ |