Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{4}\cdot12\cdot48$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 48B4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.4.509 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&19\\44&19\end{bmatrix}$, $\begin{bmatrix}17&9\\12&25\end{bmatrix}$, $\begin{bmatrix}23&0\\12&29\end{bmatrix}$, $\begin{bmatrix}23&17\\44&19\end{bmatrix}$, $\begin{bmatrix}31&28\\20&37\end{bmatrix}$, $\begin{bmatrix}41&12\\12&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.bg.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $8192$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x^{2} - y z $ |
| $=$ | $16 x y^{2} - x z^{2} - 3 w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9 x^{5} + x z^{4} + 6 y^{3} z^{2} $ |
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 |
|---|
| $(0:0:1:0)$, $(0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^8}{3^2}\cdot\frac{2277xyz^{7}w^{3}+9477xz^{2}w^{9}+65536y^{12}-256y^{2}z^{10}-5859yz^{5}w^{6}+16z^{12}-2187w^{12}}{w^{3}(xyz^{7}+81xz^{2}w^{6}+9yz^{5}w^{3}+81w^{9})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.72.4.bg.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ -9X^{5}+6Y^{3}Z^{2}+XZ^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.72.2-24.cj.1.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 48.48.0-48.g.1.31 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.14 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.7-48.eo.1.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-48.ep.1.26 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.es.1.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-48.et.1.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-48.fe.1.9 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.ff.1.33 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-48.fi.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.fj.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.8-48.hm.1.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hm.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hn.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hn.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ho.1.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ho.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hp.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hp.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hq.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hq.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hr.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hr.2.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hs.1.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hs.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ht.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ht.2.10 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hu.1.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hu.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hv.1.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hv.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hw.1.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hw.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hx.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hx.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hy.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hy.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hz.1.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.hz.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ia.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ia.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ib.1.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.ib.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.9-48.g.1.33 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.l.1.21 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.be.1.39 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.bh.1.23 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.cn.1.22 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.co.1.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cr.1.26 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cs.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.pc.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.pd.1.8 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.pg.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ph.1.20 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ps.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pt.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 48.288.9-48.pw.1.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.px.1.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 144.432.16-144.bg.1.13 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-240.bdo.1.44 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bdp.1.48 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bds.1.44 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bdt.1.48 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bee.1.36 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bef.1.52 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bei.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bej.1.36 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.xm.1.59 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xm.2.27 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xn.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xn.2.55 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xo.1.59 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xo.2.39 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xp.1.39 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xp.2.55 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xq.1.48 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xq.2.52 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xr.1.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xr.2.32 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xs.1.48 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xs.2.52 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xt.1.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xt.2.32 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xu.1.16 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xu.2.56 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xv.1.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xv.2.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xw.1.16 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xw.2.56 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xx.1.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xx.2.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xy.1.47 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xy.2.43 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xz.1.55 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xz.2.43 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ya.1.31 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ya.2.43 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yb.1.47 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yb.2.51 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.bjk.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjl.1.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjo.1.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjp.1.51 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bka.1.51 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkb.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bke.1.51 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkf.1.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.byy.1.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.byz.1.55 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzc.1.43 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzd.1.55 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzo.1.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzp.1.27 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzs.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzt.1.43 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |