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 | $6^{8}\cdot48^{2}$ | 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: | 48C8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.8.9 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&14\\44&17\end{bmatrix}$, $\begin{bmatrix}13&40\\32&1\end{bmatrix}$, $\begin{bmatrix}25&24\\24&37\end{bmatrix}$, $\begin{bmatrix}39&34\\44&9\end{bmatrix}$, $\begin{bmatrix}45&4\\32&9\end{bmatrix}$, $\begin{bmatrix}45&26\\40&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.8.p.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| 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 $ | $=$ | $ w r + u^{2} $ |
| $=$ | $w v + t^{2}$ | |
| $=$ | $x r + z u$ | |
| $=$ | $x u - z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{8} z^{3} + x^{7} y^{4} + 16 y^{8} 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 |
|---|
| $(0:0:1:0:0:0:0:0)$, $(0:1:0:0:0:0:0:0)$ |
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 24.72.4.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -v$ |
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 48.144.8.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{8}Z^{3}+X^{7}Y^{4}+16Y^{8}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.d.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.d.2.2 | $16$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 48.144.4-48.y.1.30 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.y.1.35 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bf.2.3 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bf.2.62 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-24.ch.1.21 | $48$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.15-48.ci.1.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cj.1.5 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cm.1.17 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cn.1.5 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cs.1.29 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.ct.1.5 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cu.1.3 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cv.1.5 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.17-48.ee.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ef.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eg.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eh.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ei.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ej.1.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ek.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.el.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.em.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.en.2.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eo.1.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ep.2.11 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eq.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.er.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.es.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.et.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eu.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ev.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ew.1.19 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ex.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ey.2.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ez.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.fa.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.fb.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iw.1.41 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.kf.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.oa.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ob.1.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ri.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.rj.1.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ro.2.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.rp.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |