Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 48.144.4.1315 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}21&26\\44&3\end{bmatrix}$, $\begin{bmatrix}25&15\\0&11\end{bmatrix}$, $\begin{bmatrix}29&10\\40&37\end{bmatrix}$, $\begin{bmatrix}29&43\\4&37\end{bmatrix}$, $\begin{bmatrix}29&46\\32&41\end{bmatrix}$, $\begin{bmatrix}43&45\\24&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.bh.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $8192$ |
Jacobian
| Conductor: | $2^{16}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{2}$, 576.2.a.e, 576.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x^{2} - y w $ |
| $=$ | $x y^{2} - 16 x w^{2} + 6 z^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{5} - x z^{4} + 12 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 |
|---|
| $(0:1:0:0)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{4}-16y^{2}w^{2}+16w^{4})^{3}}{w^{8}y^{2}(y-4w)(y+4w)}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.72.4.bh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{5}+12Y^{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.1.37 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 48.48.0-48.h.1.3 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.30 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
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.eq.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.er.1.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-48.eu.1.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.ev.1.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.fg.1.41 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-48.fh.1.25 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-48.fk.1.18 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-48.fl.1.19 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.8-48.ic.1.7 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ic.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.id.1.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.id.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ie.1.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ie.2.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.if.1.25 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.if.2.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ig.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ig.2.10 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ih.1.13 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ih.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ii.1.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ii.2.2 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ij.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ij.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ik.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ik.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.il.1.2 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.il.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.im.1.13 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.im.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.in.1.10 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.in.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.io.1.25 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.io.2.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ip.1.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ip.2.21 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.iq.1.21 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.iq.2.5 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ir.1.5 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.ir.2.13 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.9-48.b.1.34 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.v.1.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.z.1.33 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.bm.1.9 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.cm.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cp.1.13 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.cq.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ct.1.5 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.pe.1.3 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pf.1.18 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pi.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pj.1.9 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pu.1.7 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pv.1.6 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.py.1.7 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.pz.1.7 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 144.432.16-144.bh.1.5 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-240.bdq.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bdr.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bdu.1.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bdv.1.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.beg.1.36 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.beh.1.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bek.1.12 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.bel.1.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.yc.1.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yc.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yd.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yd.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ye.1.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ye.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yf.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yf.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yg.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yg.2.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yh.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yh.2.36 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yi.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yi.2.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yj.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yj.2.36 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yk.1.20 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yk.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yl.1.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yl.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ym.1.20 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ym.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yn.1.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yn.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yo.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yo.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yp.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yp.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yq.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yq.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yr.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.yr.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.bjm.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjn.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjq.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bjr.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkc.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkd.1.41 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkg.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bkh.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bza.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzb.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bze.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzf.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzq.1.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzr.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzu.1.27 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bzv.1.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |