Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $3^{4}\cdot6^{2}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48E8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.8.219 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&12\\36&7\end{bmatrix}$, $\begin{bmatrix}5&38\\28&35\end{bmatrix}$, $\begin{bmatrix}43&44\\20&17\end{bmatrix}$, $\begin{bmatrix}47&18\\12&29\end{bmatrix}$, $\begin{bmatrix}47&39\\24&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.8.jh.2 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: | $2^{26}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.a, 144.2.a.a, 288.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ t r - u v $ |
| $=$ | $x v - z r$ | |
| $=$ | $x t - z u$ | |
| $=$ | $x r + y v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 8 x^{6} z^{3} + 4 x^{5} y^{4} + 4 x^{4} y^{2} z^{3} + 4 x^{3} y^{6} - 2 x^{2} y^{4} z^{3} + \cdots + y^{6} z^{3} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:-1/2:1)$, $(0:0:0:0:0:0:1/2:1)$ |
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 24.72.4.gl.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 2XY+ZW $ |
| $=$ | $ 16X^{3}+2Y^{3}-4XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.144.8.jh.2 :
| $\displaystyle X$ | $=$ | $\displaystyle v+\frac{1}{2}r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{5}Y^{4}+4X^{3}Y^{6}+XY^{8}-8X^{6}Z^{3}+4X^{4}Y^{2}Z^{3}-2X^{2}Y^{4}Z^{3}+Y^{6}Z^{3} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.x.2.2 | $16$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-16.x.2.2 | $16$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.144.4-24.gl.2.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bf.1.7 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bf.1.33 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bk.1.41 | $48$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 48.144.4-48.bk.1.62 | $48$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 48.144.4-24.gl.2.32 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.15-48.xv.2.8 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.yd.2.7 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.zb.2.8 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.zj.2.7 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bbb.2.7 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bbj.1.4 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bcl.2.4 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bct.2.4 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.17-48.fb.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.fr.2.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.oh.1.22 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.op.1.23 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cjz.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ckh.2.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.clf.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cln.2.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cvr.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cvz.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cwx.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cxf.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cyh.2.8 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cyp.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.czo.1.8 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.czx.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |