Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.5.31 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&10\\40&15\end{bmatrix}$, $\begin{bmatrix}25&10\\32&39\end{bmatrix}$, $\begin{bmatrix}33&40\\16&13\end{bmatrix}$, $\begin{bmatrix}41&18\\8&43\end{bmatrix}$, $\begin{bmatrix}41&46\\8&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.5.bn.6 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{33}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 32.2.a.a, 1152.2.k.a, 1152.2.k.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} - 2 y w + z^{2} - t^{2} $ |
| $=$ | $y w + y t - z w + z t - w t - t^{2}$ | |
| $=$ | $3 x^{2} + y z - z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y + 3 x^{6} z - 5 x^{4} y^{2} z - 15 x^{4} y z^{2} + 6 x^{2} y^{3} z^{2} + 36 x^{2} y^{2} z^{3} + \cdots - 27 y z^{6} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Maps to other modular curves
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.5.bn.6 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z-\frac{1}{3}w-\frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y+3X^{6}Z-5X^{4}Y^{2}Z-15X^{4}YZ^{2}+6X^{2}Y^{3}Z^{2}+36X^{2}Y^{2}Z^{3}-2Y^{4}Z^{3}+45X^{2}YZ^{4}-18Y^{3}Z^{4}+27X^{2}Z^{5}-45Y^{2}Z^{5}-27YZ^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.1-8.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.1-8.g.2.12 | $48$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.3-48.bf.1.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bf.1.7 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.by.1.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.by.1.7 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.13-48.de.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.df.3.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dh.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dj.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dl.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dn.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dp.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dq.3.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.a.2.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.s.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.be.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.bo.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.eh.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.er.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.fh.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.fz.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.1152.37-48.bfq.6.6 | $48$ | $3$ | $3$ | $37$ | $1$ | $1^{8}\cdot2^{4}\cdot8^{2}$ |
| 48.1536.41-48.mw.5.2 | $48$ | $4$ | $4$ | $41$ | $0$ | $1^{8}\cdot2^{6}\cdot8^{2}$ |