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 | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-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: | 48.288.9.2375 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&24\\0&17\end{bmatrix}$, $\begin{bmatrix}5&8\\8&1\end{bmatrix}$, $\begin{bmatrix}11&32\\40&5\end{bmatrix}$, $\begin{bmatrix}11&42\\0&41\end{bmatrix}$, $\begin{bmatrix}33&34\\32&9\end{bmatrix}$, $\begin{bmatrix}43&0\\0&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.d.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{32}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 24.2.a.a, 32.2.a.a, 36.2.a.a$^{3}$, 48.2.a.a, 96.2.a.a, 96.2.a.b, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x y + v s + r^{2} $ |
| $=$ | $x w + x s - y t$ | |
| $=$ | $y t + z r - u v + v r$ | |
| $=$ | $x v - y z + 2 y v + w r + r s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 8 x^{6} y z^{3} + 12 x^{4} y^{4} z^{2} + 12 x^{4} y^{2} z^{4} - 6 x^{2} y^{7} z + 36 x^{2} y^{5} z^{3} + \cdots + 16 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 |
|---|
| $(0:0:0:-1:1:0:0:0:1)$, $(0:1/2:0:0:0:1:-1:1:1)$, $(0:0:0:0:1:0:0:0:0)$, $(0:-1/2:0:0:0:-1:-1:-1:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle u$ |
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.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ Y^{10}-6X^{2}Y^{7}Z+12X^{4}Y^{4}Z^{2}-5Y^{8}Z^{2}-8X^{6}YZ^{3}+36X^{2}Y^{5}Z^{3}+12X^{4}Y^{2}Z^{4}+11Y^{6}Z^{4}-6X^{2}Y^{3}Z^{5}-7Y^{4}Z^{6}-48X^{2}YZ^{7}-8Y^{2}Z^{8}+16Z^{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.v.1.31 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-48.v.1.34 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-24.ch.1.1 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.5-48.i.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 48.144.5-48.i.1.57 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
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.f.1.5 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.g.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.t.1.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.t.2.13 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.v.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.v.2.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.bl.1.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bm.1.17 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.br.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.bx.1.29 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.cj.1.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.cj.2.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.cl.1.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.cl.2.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.cy.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.dd.1.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.dp.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.dp.2.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.dr.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.dr.2.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.en.1.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.en.2.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.ep.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.17-48.ep.2.11 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.576.19-48.io.1.41 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{10}$ |
| 48.576.19-48.jk.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.jk.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.jw.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{10}$ |
| 48.576.19-48.mf.1.25 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.ml.1.3 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.ml.2.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.mn.1.21 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.oh.1.13 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{10}$ |
| 48.576.19-48.os.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.os.2.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.pb.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{10}$ |
| 48.576.19-48.pv.1.7 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.qa.1.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.qa.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{3}\cdot4$ |
| 48.576.19-48.qc.1.10 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |