Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $6^{8}\cdot12^{4}\cdot48^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48D17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.576.17.49 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&16\\8&37\end{bmatrix}$, $\begin{bmatrix}17&0\\24&37\end{bmatrix}$, $\begin{bmatrix}19&21\\0&25\end{bmatrix}$, $\begin{bmatrix}23&47\\16&41\end{bmatrix}$, $\begin{bmatrix}31&31\\8&13\end{bmatrix}$, $\begin{bmatrix}45&2\\32&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.288.17.md.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{59}\cdot3^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 32.2.a.a, 36.2.a.a$^{4}$, 72.2.d.a$^{3}$, 144.2.a.a$^{2}$, 288.2.a.a, 288.2.a.e, 288.2.d.a |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 16.192.1-16.m.2.3 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{8}\cdot2^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.1-16.m.2.3 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{8}\cdot2^{4}$ |
| 24.288.8-24.fx.1.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.bk.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.bk.1.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-24.fx.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.is.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.is.1.30 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.it.1.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.8-48.it.1.30 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.288.9-48.bf.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.288.9-48.bf.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.288.9-48.ey.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.288.9-48.ey.2.30 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.288.9-48.ez.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.288.9-48.ez.1.46 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.1152.33-48.sr.1.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.sv.1.2 | $48$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.vf.1.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.vj.1.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.zl.1.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.zp.1.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.bbx.1.2 | $48$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.1152.33-48.bcb.1.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.1152.37-48.fcl.1.2 | $48$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fct.1.2 | $48$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fcy.1.2 | $48$ | $2$ | $2$ | $37$ | $4$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fdb.1.2 | $48$ | $2$ | $2$ | $37$ | $4$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fdc.1.2 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdc.4.5 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdd.1.3 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdd.4.5 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fde.1.3 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fde.4.9 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdf.1.3 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdf.4.9 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdg.1.3 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdg.4.9 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdh.1.3 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdh.4.9 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdi.1.1 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdi.3.1 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdj.1.1 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdj.3.1 | $48$ | $2$ | $2$ | $37$ | $1$ | $2^{2}\cdot8^{2}$ |
| 48.1152.37-48.fdp.1.3 | $48$ | $2$ | $2$ | $37$ | $2$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fdr.1.3 | $48$ | $2$ | $2$ | $37$ | $2$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fdx.1.3 | $48$ | $2$ | $2$ | $37$ | $4$ | $1^{10}\cdot2^{3}\cdot4$ |
| 48.1152.37-48.fdz.1.3 | $48$ | $2$ | $2$ | $37$ | $4$ | $1^{10}\cdot2^{3}\cdot4$ |