Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| 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 $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $1^{4}\cdot2^{6}$ | ||
| 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: | 48AO9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.9.457 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}19&35\\12&31\end{bmatrix}$, $\begin{bmatrix}25&16\\24&13\end{bmatrix}$, $\begin{bmatrix}29&23\\24&19\end{bmatrix}$, $\begin{bmatrix}35&24\\12&29\end{bmatrix}$, $\begin{bmatrix}37&43\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.baz.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{39}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 32.2.a.a$^{2}$, 96.2.a.a, 96.2.a.b, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ y r - v s $ |
| $=$ | $x u + y s + z u - t u$ | |
| $=$ | $x u - y r + z u - w u$ | |
| $=$ | $x w + z w - z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} y^{4} - x^{6} y^{2} z^{2} + 12 x^{4} y^{6} + 10 x^{4} y^{4} z^{2} + 2 x^{4} y^{2} z^{4} + \cdots + 18 y^{2} z^{8} $ |
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 |
|---|
| $(0:1:0:1:-1:0:1:0:0)$, $(0:1:0:-1:1:0:1:0:0)$, $(2/3:-1:1/3:0:0:0:1:0:0)$, $(-2/3:-1:-1/3:0:0:0:1:0:0)$ |
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 24.96.3.gf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y-v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y-v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-XY^{3}-X^{2}Z^{2}-Y^{2}Z^{2}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.9.baz.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{6}Y^{4}-X^{6}Y^{2}Z^{2}+12X^{4}Y^{6}+10X^{4}Y^{4}Z^{2}+2X^{4}Y^{2}Z^{4}-16X^{2}Y^{8}-16X^{2}Y^{6}Z^{2}-14X^{2}Y^{4}Z^{4}-11X^{2}Y^{2}Z^{6}-X^{2}Z^{8}+36Y^{6}Z^{4}+54Y^{4}Z^{6}+18Y^{2}Z^{8} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.u.1.8 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.u.1.8 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.3-24.gf.2.14 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.3-48.qa.1.37 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.3-48.qa.1.62 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.5-48.mi.1.47 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.192.5-48.mi.1.61 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
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.iq.2.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.lg.1.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.ov.1.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.qu.1.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.wl.1.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.wu.1.8 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.xr.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.xy.1.8 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.zn.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.zn.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.zv.3.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.zv.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.bau.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.bau.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.bbc.2.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.bbc.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 48.768.17-48.bca.2.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bca.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bci.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bci.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bdf.3.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bdf.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bdn.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.768.17-48.bdn.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 48.1152.33-48.bpg.2.24 | $48$ | $3$ | $3$ | $33$ | $2$ | $1^{12}\cdot2^{6}$ |