Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $1^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48E9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.9.2380 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&46\\32&9\end{bmatrix}$, $\begin{bmatrix}17&4\\8&29\end{bmatrix}$, $\begin{bmatrix}19&20\\16&25\end{bmatrix}$, $\begin{bmatrix}39&10\\32&33\end{bmatrix}$, $\begin{bmatrix}41&10\\16&25\end{bmatrix}$, $\begin{bmatrix}47&16\\16&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.e.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{40}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 64.2.a.a, 144.2.a.a, 192.2.a.a, 192.2.a.b, 192.2.a.c, 192.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x y - u s + v^{2} $ |
| $=$ | $x t + x s - y z$ | |
| $=$ | $y z + w v + u v + u r$ | |
| $=$ | $x w + x u - y w - 2 y u + r s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{6} z^{7} + 12 x^{4} y^{3} z^{6} - 12 x^{4} y z^{8} - 48 x^{2} y^{8} z^{3} + 6 x^{2} y^{6} z^{5} + \cdots - y^{3} z^{10} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:0:-1:0:0:0:1)$, $(0:0:0:-2:-2:1:0:0:0)$, $(0:0:1:0:0:0:0:0:0)$, $(0:0:0:-2:2:1:0:0:0)$ |
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 24.72.4.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 4Y^{2}+ZW $ |
| $=$ | $ X^{3}+YZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.144.9.e.2 :
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 16Y^{13}+8Y^{11}Z^{2}-48X^{2}Y^{8}Z^{3}-7Y^{9}Z^{4}+6X^{2}Y^{6}Z^{5}+12X^{4}Y^{3}Z^{6}-11Y^{7}Z^{6}+8X^{6}Z^{7}+36X^{2}Y^{4}Z^{7}-12X^{4}YZ^{8}-5Y^{5}Z^{8}+6X^{2}Y^{2}Z^{9}-Y^{3}Z^{10} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-48.w.1.13 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-48.w.1.20 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-24.ch.1.24 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.5-48.l.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.144.5-48.l.1.29 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.17-48.e.2.11 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.i.2.11 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.x.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.x.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.bk.2.17 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bo.2.11 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bq.2.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.bz.2.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.cn.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.cn.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.cz.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.df.2.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.dt.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.dt.2.15 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.er.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.er.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.19-48.io.1.41 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{10}$ |
| 48.576.19-48.jn.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.jn.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.jq.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.jq.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ki.1.26 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mb.2.13 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mo.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mo.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mq.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mq.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.na.2.9 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.og.2.12 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.ov.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ov.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ox.1.28 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ox.2.26 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.pj.2.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{10}$ |
| 48.576.19-48.pr.2.2 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.qd.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.qd.2.4 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.qf.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.qf.2.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.qq.2.1 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |