Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $8$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $1^{8}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AK9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.9.8588 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&34\\0&13\end{bmatrix}$, $\begin{bmatrix}5&40\\0&5\end{bmatrix}$, $\begin{bmatrix}11&28\\24&41\end{bmatrix}$, $\begin{bmatrix}25&10\\24&5\end{bmatrix}$, $\begin{bmatrix}41&10\\0&1\end{bmatrix}$, $\begin{bmatrix}41&26\\24&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.hr.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: | $2^{37}\cdot3^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 24.2.a.a$^{2}$, 48.2.a.a, 72.2.a.a, 144.2.a.b, 288.2.a.b, 288.2.a.c, 288.2.a.d$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ t v - t r - u s $ |
| $=$ | $z s - w r - w s$ | |
| $=$ | $x y + x t - z s$ | |
| $=$ | $x v - y w - w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{11} + 9 x^{9} z^{2} + 2 x^{8} y^{2} z + 6 x^{7} z^{4} + 24 x^{6} y^{2} z^{3} - 6 x^{5} z^{6} + \cdots + 2 y^{2} z^{9} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:1:0)$, $(0:-2/3:0:0:-1/3:-1/3:-1:-2:1)$, $(0:0:0:0:0:0:-1:-1:1)$, $(0:1/2:0:0:-1/2:1:0:0:0)$, $(0:0:0:0:0:1:1:0:0)$, $(0:-1:0:0:1:1:0:0:0)$, $(0:0:0:0:0:-1:1:0:0)$, $(0:2/3:0:0:1/3:1/3:-1:-2:1)$ |
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 48.96.5.os.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z-w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
| $\displaystyle W$ | $=$ | $\displaystyle v+s$ |
| $\displaystyle T$ | $=$ | $\displaystyle -v+r+s$ |
Equation of the image curve:
| $0$ | $=$ | $ XW-ZW+ZT $ |
| $=$ | $ X^{2}+XZ-2Z^{2}-W^{2}-WT $ | |
| $=$ | $ 6Y^{2}+2XW+ZW+XT-ZT $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.9.hr.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{11}+9X^{9}Z^{2}+2X^{8}Y^{2}Z+6X^{7}Z^{4}+24X^{6}Y^{2}Z^{3}-6X^{5}Z^{6}-20X^{4}Y^{2}Z^{5}+12X^{3}Y^{4}Z^{4}-9X^{3}Z^{8}+24X^{2}Y^{2}Z^{7}-12XY^{4}Z^{6}-3XZ^{10}+2Y^{2}Z^{9} $ |
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.cl.1.18 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 48.96.1-48.b.1.2 | $48$ | $4$ | $4$ | $1$ | $0$ | $1^{8}$ |
| 48.192.3-24.cl.1.12 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 48.192.3-48.qh.1.10 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 48.192.3-48.qh.1.23 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 48.192.5-48.os.1.10 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5-48.os.1.23 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.17-48.gv.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.gv.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.gv.3.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.gv.6.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.hd.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.hd.2.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.hd.3.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.hd.4.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.hf.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.hf.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.hf.3.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.hf.6.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.hw.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.hw.2.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.io.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.io.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.21-48.et.1.41 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{12}$ |
| 48.768.21-48.fo.1.19 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.fo.2.5 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.fr.1.18 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.fr.2.8 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.fz.1.20 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{12}$ |
| 48.768.21-48.je.1.17 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{12}$ |
| 48.768.21-48.jv.1.9 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.jv.2.2 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.jy.1.18 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.jy.2.3 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4$ |
| 48.768.21-48.kd.1.18 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{12}$ |
| 48.768.21-48.ko.1.1 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.ko.2.1 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.ko.3.1 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.ko.4.1 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lc.3.13 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lc.4.13 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lm.3.7 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lm.4.7 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lq.1.5 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lq.2.5 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lq.3.9 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.768.21-48.lq.4.9 | $48$ | $2$ | $2$ | $21$ | $1$ | $2^{2}\cdot4^{2}$ |
| 48.1152.33-48.k.1.7 | $48$ | $3$ | $3$ | $33$ | $3$ | $1^{24}$ |