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 | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 48.288.8.350 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&2\\44&17\end{bmatrix}$, $\begin{bmatrix}27&26\\28&9\end{bmatrix}$, $\begin{bmatrix}37&2\\40&1\end{bmatrix}$, $\begin{bmatrix}37&20\\40&1\end{bmatrix}$, $\begin{bmatrix}37&32\\8&5\end{bmatrix}$, $\begin{bmatrix}47&42\\12&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.8.n.1 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: | $2^{26}\cdot3^{16}$ |
| 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 $ | $=$ | $ z v + w r $ |
| $=$ | $x v - y w$ | |
| $=$ | $x r + y z$ | |
| $=$ | $x v + y w + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 48 x^{8} y^{3} - x^{7} z^{4} + 243 y^{3} z^{8} $ |
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 |
|---|
| $(0:0:0:1:0:0:0:0)$, $(0:0:1:0:0:0:0:0)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -u$ |
| $\displaystyle W$ | $=$ | $\displaystyle t$ |
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.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ | $=$ | $ -48X^{8}Y^{3}-X^{7}Z^{4}+243Y^{3}Z^{8} $ |
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 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 48.96.0-48.d.1.2 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.144.4-48.z.2.5 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.z.2.60 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.be.1.30 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.be.1.35 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-24.ch.1.5 | $48$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
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.bw.1.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bx.2.9 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.ca.2.30 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cb.1.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cc.2.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cd.1.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.ce.1.19 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cf.2.5 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.17-48.dg.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dh.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.di.1.18 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dj.2.18 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dk.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dl.2.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dm.2.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dn.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.do.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dp.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dq.2.11 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dr.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ds.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dt.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.du.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dv.1.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dw.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dx.1.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dy.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dz.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ea.1.37 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eb.2.19 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ec.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ed.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.is.1.41 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jb.1.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jy.2.26 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ke.2.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ra.2.12 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.rb.1.4 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.rc.1.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.rd.1.7 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |