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: | $24$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.5.25 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&2\\40&39\end{bmatrix}$, $\begin{bmatrix}1&22\\8&39\end{bmatrix}$, $\begin{bmatrix}1&46\\16&15\end{bmatrix}$, $\begin{bmatrix}9&44\\16&45\end{bmatrix}$, $\begin{bmatrix}33&14\\16&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.5.cw.2 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: | no |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 32.2.a.a, 1152.2.k.a$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y t - z w $ |
| $=$ | $y^{2} + 2 y t + z^{2} + 2 z w - w^{2} - t^{2}$ | |
| $=$ | $6 x^{2} + y t + z w - w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} - 2 x^{4} y^{2} z^{2} + x^{4} z^{4} - 12 x^{2} y^{4} z^{2} - 12 x^{2} y^{2} z^{4} + \cdots + 9 y^{2} z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=31$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 48.96.3.by.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2x+z-w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+w$ |
Equation of the image curve:
| $0$ | $=$ | $ 10X^{4}-8X^{3}Y-2X^{2}Y^{2}+XY^{3}+4X^{3}Z+4X^{2}YZ+4XY^{2}Z-Y^{3}Z-2X^{2}Z^{2}-2XYZ^{2}-5Y^{2}Z^{2}-4XZ^{3}-6YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.5.cw.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}-2X^{4}Y^{2}Z^{2}+X^{4}Z^{4}-12X^{2}Y^{4}Z^{2}-12X^{2}Y^{2}Z^{4}+9Y^{6}Z^{2}+18Y^{4}Z^{4}+9Y^{2}Z^{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.2.14 | $48$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.2-48.c.1.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.c.1.27 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.g.2.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.g.2.28 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.3-48.by.1.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.by.1.31 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
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.dx.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.dz.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ea.1.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.eb.2.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ec.2.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ed.2.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ee.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ef.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.eh.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.ei.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.ej.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.ek.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.1152.37-48.bvj.1.2 | $48$ | $3$ | $3$ | $37$ | $1$ | $1^{8}\cdot2^{4}\cdot8^{2}$ |
| 48.1536.41-48.tu.1.6 | $48$ | $4$ | $4$ | $41$ | $0$ | $1^{8}\cdot2^{6}\cdot8^{2}$ |