Invariants
| Level: | $228$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-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: | $\begin{bmatrix}41&106\\146&117\end{bmatrix}$, $\begin{bmatrix}65&100\\50&87\end{bmatrix}$, $\begin{bmatrix}67&26\\126&53\end{bmatrix}$, $\begin{bmatrix}209&120\\68&91\end{bmatrix}$, $\begin{bmatrix}213&94\\130&177\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 228.48.1.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: | $5909760$ |
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 |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 76.12.0.a.1 | $76$ | $8$ | $4$ | $0$ | $?$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 228.48.0-6.a.1.8 | $228$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 228.48.0-228.n.1.2 | $228$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 228.48.0-228.n.1.15 | $228$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 228.48.1-228.p.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.48.1-228.p.1.16 | $228$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 228.192.1-228.e.1.2 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.e.2.2 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.e.3.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.e.4.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.3-228.a.1.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.c.1.4 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.g.1.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.i.1.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.o.1.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.o.2.6 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.s.1.6 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.s.2.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.288.5-228.a.1.2 | $228$ | $3$ | $3$ | $5$ | $?$ | not computed |