Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{16}\cdot16^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.5.640 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&1\\0&35\end{bmatrix}$, $\begin{bmatrix}17&1\\32&35\end{bmatrix}$, $\begin{bmatrix}27&40\\16&23\end{bmatrix}$, $\begin{bmatrix}45&11\\40&47\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.384.5-48.hw.1.1, 48.384.5-48.hw.1.2, 48.384.5-48.hw.1.3, 48.384.5-48.hw.1.4, 48.384.5-48.hw.1.5, 48.384.5-48.hw.1.6, 48.384.5-48.hw.1.7, 48.384.5-48.hw.1.8, 240.384.5-48.hw.1.1, 240.384.5-48.hw.1.2, 240.384.5-48.hw.1.3, 240.384.5-48.hw.1.4, 240.384.5-48.hw.1.5, 240.384.5-48.hw.1.6, 240.384.5-48.hw.1.7, 240.384.5-48.hw.1.8 |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{30}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 64.2.a.a, 576.2.a.c$^{2}$, 576.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - x y + y^{2} + 2 w^{2} $ |
| $=$ | $2 x^{2} - x y - 2 x z + y^{2} - 2 z^{2} - 2 w^{2}$ | |
| $=$ | $x^{2} - 2 x y + 2 x z - 4 y z - t^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1.n.1 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.96.1.cv.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 48.96.1.bp.2 | $48$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 48.96.3.fx.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.96.3.fy.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.fz.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.gj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.37.fnl.2 | $48$ | $3$ | $3$ | $37$ | $5$ | $1^{16}\cdot2^{6}\cdot4$ |
| 48.768.41.bwl.2 | $48$ | $4$ | $4$ | $41$ | $7$ | $1^{18}\cdot2^{7}\cdot4$ |