Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| 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: | 48.288.9.2381 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&26\\8&5\end{bmatrix}$, $\begin{bmatrix}5&18\\0&41\end{bmatrix}$, $\begin{bmatrix}7&0\\0&41\end{bmatrix}$, $\begin{bmatrix}9&14\\40&33\end{bmatrix}$, $\begin{bmatrix}17&24\\24&1\end{bmatrix}$, $\begin{bmatrix}35&32\\16&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.c.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: | $2^{32}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.a.a, 144.2.a.a, 144.2.a.b, 288.2.a.b, 288.2.a.c, 288.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x u - x s - y v $ |
| $=$ | $y v - z t - z r + w r$ | |
| $=$ | $t^{2} - u^{2} + 2 u s - v s$ | |
| $=$ | $y u + y v - 2 y s + w r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 27 x^{6} z^{7} + 27 x^{4} y^{3} z^{6} - 27 x^{4} y z^{8} + 72 x^{2} y^{8} z^{3} - 9 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 |
|---|
| $(0:0:0:0:0:1:1:0:1)$, $(0:0:0:0:-2:-2:0:1:0)$, $(0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:-2:2:0:1:0)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w$ |
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.c.2 :
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16Y^{13}+8Y^{11}Z^{2}+72X^{2}Y^{8}Z^{3}-7Y^{9}Z^{4}-9X^{2}Y^{6}Z^{5}+27X^{4}Y^{3}Z^{6}-11Y^{7}Z^{6}-27X^{6}Z^{7}-54X^{2}Y^{4}Z^{7}-27X^{4}YZ^{8}-5Y^{5}Z^{8}-9X^{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.t.1.13 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-48.t.1.20 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-24.ch.1.16 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.5-48.o.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.144.5-48.o.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.d.2.6 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.h.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.r.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.r.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.bi.2.6 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bn.2.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.bs.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bv.2.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.ch.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.ch.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.da.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.dc.2.11 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.dn.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.dn.2.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.el.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.el.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.19-48.ik.2.33 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.jf.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.jf.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ji.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ji.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.jr.1.26 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mb.1.18 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mh.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mh.2.7 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mj.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mj.2.7 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.mn.1.21 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.oj.2.4 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.oo.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.oo.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.oq.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.oq.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ou.2.12 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.pp.2.2 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.pw.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.pw.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.py.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.py.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.qg.2.2 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |