Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $72$ | ||
| 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: | 48.144.4.1077 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&13\\40&11\end{bmatrix}$, $\begin{bmatrix}25&26\\44&11\end{bmatrix}$, $\begin{bmatrix}25&44\\44&23\end{bmatrix}$, $\begin{bmatrix}33&40\\40&5\end{bmatrix}$, $\begin{bmatrix}35&22\\0&19\end{bmatrix}$, $\begin{bmatrix}45&46\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.z.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: | $2^{10}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.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} + 2 x^{4} y^{2} + 9 x^{2} y z^{3} + 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.z.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{6}+2X^{4}Y^{2}+9X^{2}YZ^{3}+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.1.20 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
| 48.48.0-48.e.2.1 | $48$ | $3$ | $3$ | $0$ | $0$ | 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.dh.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dl.2.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2$ |
| 48.288.7-48.dp.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dt.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dy.1.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.ed.1.19 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.eh.2.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.el.1.23 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.8-48.n.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.r.1.14 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.bl.1.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.bu.2.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.df.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.dg.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dk.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.dr.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.du.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dv.1.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ek.1.21 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.el.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fa.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fb.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fq.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fr.1.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gf.1.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gj.1.10 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gn.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gr.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gv.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.gz.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hd.1.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.hh.1.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ho.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hp.1.21 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ie.1.19 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.if.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.iu.1.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.iv.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.jk.1.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.jl.1.5 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.9-48.ek.2.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.el.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.fa.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.fb.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.is.2.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.it.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.ji.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.jj.2.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.mo.1.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.mp.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.ne.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.nf.1.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.ny.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.nz.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oo.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.op.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 144.432.16-144.r.1.25 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-240.vx.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wf.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wn.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wv.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xd.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xl.1.18 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xt.2.12 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.yb.2.38 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.er.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ez.2.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fh.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fp.2.13 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gf.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gn.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gv.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.hd.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.jy.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.jz.1.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.le.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.lf.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.mk.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ml.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nq.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nr.1.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ov.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pd.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pl.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pt.2.2 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qb.1.14 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qj.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qr.1.22 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qz.1.20 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.sq.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.sr.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tw.1.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tx.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.vc.1.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.vd.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wi.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wj.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.xe.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.xf.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.yk.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.yl.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zq.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zr.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.baw.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bax.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bms.1.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmt.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bny.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bnz.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bpe.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bpf.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bqk.1.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bql.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |