Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| 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 | $3^{2}\cdot6^{3}\cdot48$ | 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: | 48D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.4.1126 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&21\\20&29\end{bmatrix}$, $\begin{bmatrix}3&35\\20&31\end{bmatrix}$, $\begin{bmatrix}11&23\\8&9\end{bmatrix}$, $\begin{bmatrix}15&19\\16&41\end{bmatrix}$, $\begin{bmatrix}23&19\\4&43\end{bmatrix}$, $\begin{bmatrix}33&38\\16&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.be.2 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: | $2^{14}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2$ |
| Newforms: | 36.2.a.a$^{2}$, 288.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x z + y w $ |
| $=$ | $48 x^{3} - x y^{2} + 6 z^{3} - 4 z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{6} - 9 x^{4} z^{2} + 2 x^{2} y^{3} z - 6 y^{3} 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 |
|---|
| $(0:1:0:0)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{3145725x^{2}y^{10}-452967552x^{2}y^{7}w^{3}+920205312x^{2}y^{4}w^{6}-21430272x^{2}yw^{9}-65536y^{12}-9437187y^{9}z^{2}w+19922942y^{9}w^{3}+176178144y^{6}z^{2}w^{4}-154128960y^{6}w^{6}-36661248y^{3}z^{2}w^{7}+25133056y^{3}w^{9}-1474560z^{2}w^{10}-65536w^{12}}{w^{3}y(3x^{2}y^{6}+288x^{2}y^{3}w^{3}-3072x^{2}w^{6}+3y^{5}z^{2}w+2y^{5}w^{3}+192y^{2}z^{2}w^{4})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.72.4.be.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{6}+2X^{2}Y^{3}Z-9X^{4}Z^{2}-6Y^{3}Z^{3} $ |
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.1.18 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
| 48.48.0-48.f.1.1 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.30 | $48$ | $2$ | $2$ | $2$ | $0$ | $2$ |
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.di.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dm.1.3 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2$ |
| 48.288.7-48.dq.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.du.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dz.2.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.ee.2.21 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.ei.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.em.2.21 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.8-48.n.2.33 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.w.1.10 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.bq.1.30 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.bu.1.22 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dc.2.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.dj.2.2 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dn.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.do.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ee.1.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ef.2.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.eu.2.19 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ev.2.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fk.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fl.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ga.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gb.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gg.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gk.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.go.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gs.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gw.2.10 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ha.2.10 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.he.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.hi.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hy.2.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hz.2.19 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.io.2.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ip.1.17 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.je.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.jf.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ju.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.jv.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.9-48.eu.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.ev.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.fk.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.fl.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.jc.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.jd.1.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.js.1.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.jt.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.my.2.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.mz.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.no.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.np.2.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oi.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oj.2.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oy.2.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oz.2.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 144.432.16-144.w.2.17 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-240.wc.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wk.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.ws.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xa.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xi.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xq.2.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xy.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.yg.2.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.ew.1.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fe.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fm.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fu.1.39 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gk.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gs.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ha.1.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.hi.1.19 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ky.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.kz.2.9 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.me.2.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.mf.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nk.2.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nl.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.oq.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.or.2.9 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pa.2.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pi.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pq.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.py.1.2 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qg.1.14 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qo.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qw.2.22 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.re.2.20 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tq.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tr.2.19 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.uw.2.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ux.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wc.2.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wd.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xi.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.xj.2.19 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.ye.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.yf.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zk.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zl.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.baq.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bar.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bbw.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bbx.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bns.1.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bnt.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.boy.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.boz.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bqe.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bqf.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.brk.1.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.brl.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |