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: | 48.144.4.442 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&21\\12&7\end{bmatrix}$, $\begin{bmatrix}11&39\\44&43\end{bmatrix}$, $\begin{bmatrix}27&4\\16&43\end{bmatrix}$, $\begin{bmatrix}33&2\\44&35\end{bmatrix}$, $\begin{bmatrix}41&1\\16&27\end{bmatrix}$, $\begin{bmatrix}41&29\\20&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.z.2 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: | $1^{2}\cdot2$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x z - y w $ |
| $=$ | $24 x^{3} + x y^{2} + 3 z^{3} + 4 z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{6} + x^{4} y^{2} + 18 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{3145725x^{2}y^{10}-452967552x^{2}y^{7}w^{3}+920205312x^{2}y^{4}w^{6}-21430272x^{2}yw^{9}+131072y^{12}-9437187y^{9}z^{2}w-39845884y^{9}w^{3}+176178144y^{6}z^{2}w^{4}+308257920y^{6}w^{6}-36661248y^{3}z^{2}w^{7}-50266112y^{3}w^{9}-1474560z^{2}w^{10}+131072w^{12}}{w^{3}y(3x^{2}y^{6}+288x^{2}y^{3}w^{3}-3072x^{2}w^{6}+3y^{5}z^{2}w-4y^{5}w^{3}+192y^{2}z^{2}w^{4})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.72.4.z.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{6}+X^{4}Y^{2}+18X^{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.1.11 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
| 48.48.0-48.e.1.29 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.14 | $48$ | $2$ | $2$ | $2$ | $0$ | $2$ |
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.1.24 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dl.1.22 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2$ |
| 48.288.7-48.dp.2.12 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dt.2.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.dy.2.21 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.ed.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.eh.1.14 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.7-48.el.2.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 48.288.8-48.n.1.33 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.r.1.16 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.bl.1.32 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.bu.1.8 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.df.2.19 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.dg.2.20 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dk.1.17 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.dr.1.18 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.du.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.dv.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.ek.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.el.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fa.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fb.2.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fq.2.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.fr.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gf.2.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gj.2.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gn.1.23 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gr.1.24 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.gv.2.14 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.gz.1.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hd.2.14 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.hh.2.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ho.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.hp.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.ie.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.if.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.iu.2.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.iv.2.11 | $48$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 48.288.8-48.jk.2.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.8-48.jl.2.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 48.288.9-48.ek.1.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.el.1.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.fa.1.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.fb.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.is.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.it.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.ji.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.jj.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.mo.2.6 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.mp.1.8 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 48.288.9-48.ne.1.8 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.nf.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.ny.1.12 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.nz.2.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.oo.2.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 48.288.9-48.op.1.12 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 144.432.16-144.r.2.25 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-240.vx.1.48 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wf.1.48 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wn.2.16 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.wv.2.16 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xd.2.12 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xl.2.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.xt.1.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.yb.1.16 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.er.1.50 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ez.1.32 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fh.2.38 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.fp.1.16 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gf.2.37 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gn.2.39 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.gv.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.hd.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.jy.2.57 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.jz.2.41 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.le.1.37 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.lf.2.45 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.mk.1.21 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ml.2.29 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nq.2.57 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.nr.2.41 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ov.2.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pd.2.32 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pl.1.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.pt.1.48 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qb.2.42 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qj.2.36 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qr.2.58 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.qz.2.52 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.sq.2.47 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.sr.1.39 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tw.2.43 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.tx.2.59 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.vc.2.43 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.vd.2.59 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wi.2.31 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.wj.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.xe.1.27 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.xf.1.47 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.yk.1.59 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.yl.2.51 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zq.1.59 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.zr.2.51 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.baw.1.43 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bax.1.55 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bms.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmt.1.58 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bny.1.30 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bnz.1.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bpe.1.46 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bpf.1.42 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bqk.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bql.1.58 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |