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: | 48.288.17.4232 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&23\\8&3\end{bmatrix}$, $\begin{bmatrix}13&42\\32&5\end{bmatrix}$, $\begin{bmatrix}15&19\\16&17\end{bmatrix}$, $\begin{bmatrix}17&24\\4&43\end{bmatrix}$, $\begin{bmatrix}33&32\\28&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.576.17-48.gs.1.1, 48.576.17-48.gs.1.2, 48.576.17-48.gs.1.3, 48.576.17-48.gs.1.4, 48.576.17-48.gs.1.5, 48.576.17-48.gs.1.6, 48.576.17-48.gs.1.7, 48.576.17-48.gs.1.8, 48.576.17-48.gs.1.9, 48.576.17-48.gs.1.10, 48.576.17-48.gs.1.11, 48.576.17-48.gs.1.12, 48.576.17-48.gs.1.13, 48.576.17-48.gs.1.14, 48.576.17-48.gs.1.15, 48.576.17-48.gs.1.16 |
| Cyclic 48-isogeny field degree: | $8$ |
| 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 $\Q_p$ 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.ft.1 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.96.1.w.2 | $48$ | $3$ | $3$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.144.8.w.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.144.8.hy.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.144.8.hz.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.144.9.l.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 48.144.9.jc.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.144.9.jd.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
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.ls.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.lu.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.ma.1 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.mc.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.mz.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.nc.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.nl.1 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.576.33.no.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2^{4}$ |