Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{4}\cdot48^{4}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48D17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.17.4522 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&43\\16&11\end{bmatrix}$, $\begin{bmatrix}13&41\\16&19\end{bmatrix}$, $\begin{bmatrix}23&10\\32&5\end{bmatrix}$, $\begin{bmatrix}41&3\\0&1\end{bmatrix}$, $\begin{bmatrix}47&45\\0&29\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.576.17-48.clm.2.1, 48.576.17-48.clm.2.2, 48.576.17-48.clm.2.3, 48.576.17-48.clm.2.4, 48.576.17-48.clm.2.5, 48.576.17-48.clm.2.6, 48.576.17-48.clm.2.7, 48.576.17-48.clm.2.8, 48.576.17-48.clm.2.9, 48.576.17-48.clm.2.10, 48.576.17-48.clm.2.11, 48.576.17-48.clm.2.12, 48.576.17-48.clm.2.13, 48.576.17-48.clm.2.14, 48.576.17-48.clm.2.15, 48.576.17-48.clm.2.16 |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{76}\cdot3^{34}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.a, 144.2.a.a, 288.2.d.a$^{3}$, 576.2.a.a, 576.2.a.c, 576.2.a.e, 576.2.a.f, 576.2.a.i |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31,127$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.8.gs.1 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.96.1.eh.1 | $48$ | $3$ | $3$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.144.8.dr.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.144.8.iu.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.144.8.jg.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.144.9.is.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.144.9.je.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.144.9.lx.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.33.cft.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cfx.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cgz.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.chd.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cvx.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cwb.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cxk.1 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.cxp.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |