Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}19&81\\28&131\end{bmatrix}$, $\begin{bmatrix}29&88\\20&63\end{bmatrix}$, $\begin{bmatrix}127&26\\60&125\end{bmatrix}$, $\begin{bmatrix}143&112\\140&57\end{bmatrix}$, $\begin{bmatrix}177&133\\112&117\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 228.96.1-228.l.1.1, 228.96.1-228.l.1.2, 228.96.1-228.l.1.3, 228.96.1-228.l.1.4, 228.96.1-228.l.1.5, 228.96.1-228.l.1.6, 228.96.1-228.l.1.7, 228.96.1-228.l.1.8, 228.96.1-228.l.1.9, 228.96.1-228.l.1.10, 228.96.1-228.l.1.11, 228.96.1-228.l.1.12 |
| Cyclic 228-isogeny field degree: | $20$ |
| Cyclic 228-torsion field degree: | $1440$ |
| Full 228-torsion field degree: | $11819520$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 76.12.0.h.1 | $76$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(12)$ | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 76.12.0.h.1 | $76$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
| 228.24.0.m.1 | $228$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 228.24.1.p.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 228.96.1.m.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1.m.2 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1.m.3 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1.m.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.144.5.cb.1 | $228$ | $3$ | $3$ | $5$ | $?$ | not computed |