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: | $2$ | ||||||
| $\Q$-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: | 48.288.9.2382 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&22\\16&9\end{bmatrix}$, $\begin{bmatrix}13&14\\32&5\end{bmatrix}$, $\begin{bmatrix}15&44\\32&45\end{bmatrix}$, $\begin{bmatrix}37&32\\16&1\end{bmatrix}$, $\begin{bmatrix}41&12\\24&1\end{bmatrix}$, $\begin{bmatrix}41&20\\16&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.f.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: | $2^{40}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 576.2.a.b, 576.2.a.c, 576.2.a.d, 576.2.a.g, 576.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ t^{2} - u^{2} + u r + v r $ |
| $=$ | $y u - 2 y r - w s$ | |
| $=$ | $x v + x r + z s + w t + w s$ | |
| $=$ | $x r + y u + y v + w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{13} - 2 x^{11} y^{2} - 3 x^{9} y^{4} + 18 x^{8} y^{3} z^{2} - 4 x^{7} y^{6} + 18 x^{6} y^{5} z^{2} + \cdots + 54 y^{7} z^{6} $ |
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 |
|---|
| $(0:0:0:0:0:1:0:1:0)$, $(0:0:0:0:-2:-2:-2:0:1)$, $(0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:-2:2:2:0:1)$ |
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 24.72.4.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -z$ |
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.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{13}-2X^{11}Y^{2}-3X^{9}Y^{4}+18X^{8}Y^{3}Z^{2}-4X^{7}Y^{6}+18X^{6}Y^{5}Z^{2}-3X^{5}Y^{8}-18X^{4}Y^{7}Z^{2}-2X^{3}Y^{10}-18X^{2}Y^{9}Z^{2}-54X^{2}Y^{5}Z^{6}-XY^{12}+54Y^{7}Z^{6} $ |
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.s.1.15 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 48.144.4-48.s.1.18 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 48.144.4-24.ch.1.8 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.5-48.p.1.8 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.144.5-48.p.1.25 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
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.c.2.6 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.j.2.3 | $48$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 48.576.17-48.ba.1.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.ba.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.bj.2.18 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bp.2.3 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.bu.2.29 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.by.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.cq.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.cq.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.db.2.6 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.de.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.dw.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.dw.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.eu.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.17-48.eu.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.19-48.ik.1.41 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.js.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.js.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.jv.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.jv.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.kj.1.21 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.mf.1.25 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mr.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.mr.2.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.mt.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.mt.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.na.1.21 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.on.2.4 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.oy.1.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.oy.2.8 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.pa.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.pa.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.pi.2.3 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.po.2.2 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.qh.1.1 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.qh.2.2 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.qj.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.qj.2.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.qr.2.1 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |