Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{6}\cdot4^{2}$ | ||
| 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: | 48AP7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.7.2158 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&13\\0&23\end{bmatrix}$, $\begin{bmatrix}11&33\\12&43\end{bmatrix}$, $\begin{bmatrix}29&35\\0&11\end{bmatrix}$, $\begin{bmatrix}35&15\\36&47\end{bmatrix}$, $\begin{bmatrix}43&5\\12&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.gt.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{30}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 72.2.a.a, 96.2.d.a, 144.2.a.b, 288.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t + z u + w t $ |
| $=$ | $x y - y w - z v$ | |
| $=$ | $2 z t - w u + w v$ | |
| $=$ | $x u + x v + 2 y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{4} y^{4} - 18 x^{4} y^{2} z^{2} - x^{2} y^{6} + 21 x^{2} y^{4} z^{2} + 9 x^{2} y^{2} z^{4} + \cdots + 6 y^{4} z^{4} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, 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 24.96.3.gf.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{2}+2Y^{4}-X^{3}Z+Y^{2}Z^{2}+XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.7.gt.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{4}Y^{4}-18X^{4}Y^{2}Z^{2}-X^{2}Y^{6}+21X^{2}Y^{4}Z^{2}+9X^{2}Y^{2}Z^{4}+27X^{2}Z^{6}-2Y^{6}Z^{2}+6Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.gf.1.18 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-24.gf.1.3 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.96.0-48.bg.2.11 | $48$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 48.192.3-48.qc.1.37 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qc.1.43 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qf.1.20 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.qf.1.56 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
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.ll.3.12 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.ll.4.12 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.lt.1.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.lt.2.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.ms.3.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.ms.4.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.na.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.na.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.17-48.hv.1.18 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.jy.1.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.po.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.qd.2.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.wi.1.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.wl.2.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.xm.2.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.xv.2.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bnw.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bnw.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.boe.3.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.boe.4.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bpb.1.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bpb.2.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bpj.3.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bpj.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.1152.29-48.hh.1.4 | $48$ | $3$ | $3$ | $29$ | $0$ | $1^{10}\cdot2^{6}$ |