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Properties

Label 48.72.4.bg.1
Level $48$
Index $72$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $144$
Index: $72$ $\PSL_2$-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: 48.72.4.66

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&25\\32&47\end{bmatrix}$, $\begin{bmatrix}5&26\\8&37\end{bmatrix}$, $\begin{bmatrix}11&38\\32&7\end{bmatrix}$, $\begin{bmatrix}25&21\\24&19\end{bmatrix}$, $\begin{bmatrix}31&1\\16&29\end{bmatrix}$, $\begin{bmatrix}39&37\\40&45\end{bmatrix}$, $\begin{bmatrix}39&47\\20&39\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.144.4-48.bg.1.1, 48.144.4-48.bg.1.2, 48.144.4-48.bg.1.3, 48.144.4-48.bg.1.4, 48.144.4-48.bg.1.5, 48.144.4-48.bg.1.6, 48.144.4-48.bg.1.7, 48.144.4-48.bg.1.8, 48.144.4-48.bg.1.9, 48.144.4-48.bg.1.10, 48.144.4-48.bg.1.11, 48.144.4-48.bg.1.12, 48.144.4-48.bg.1.13, 48.144.4-48.bg.1.14, 48.144.4-48.bg.1.15, 48.144.4-48.bg.1.16, 48.144.4-48.bg.1.17, 48.144.4-48.bg.1.18, 48.144.4-48.bg.1.19, 48.144.4-48.bg.1.20, 48.144.4-48.bg.1.21, 48.144.4-48.bg.1.22, 48.144.4-48.bg.1.23, 48.144.4-48.bg.1.24, 48.144.4-48.bg.1.25, 48.144.4-48.bg.1.26, 48.144.4-48.bg.1.27, 48.144.4-48.bg.1.28, 48.144.4-48.bg.1.29, 48.144.4-48.bg.1.30, 48.144.4-48.bg.1.31, 48.144.4-48.bg.1.32, 48.144.4-48.bg.1.33, 48.144.4-48.bg.1.34, 48.144.4-48.bg.1.35, 48.144.4-48.bg.1.36, 48.144.4-48.bg.1.37, 48.144.4-48.bg.1.38, 48.144.4-48.bg.1.39, 48.144.4-48.bg.1.40, 48.144.4-48.bg.1.41, 48.144.4-48.bg.1.42, 48.144.4-48.bg.1.43, 48.144.4-48.bg.1.44, 48.144.4-48.bg.1.45, 48.144.4-48.bg.1.46, 48.144.4-48.bg.1.47, 48.144.4-48.bg.1.48, 48.144.4-48.bg.1.49, 48.144.4-48.bg.1.50, 48.144.4-48.bg.1.51, 48.144.4-48.bg.1.52, 48.144.4-48.bg.1.53, 48.144.4-48.bg.1.54, 48.144.4-48.bg.1.55, 48.144.4-48.bg.1.56, 48.144.4-48.bg.1.57, 48.144.4-48.bg.1.58, 48.144.4-48.bg.1.59, 48.144.4-48.bg.1.60, 48.144.4-48.bg.1.61, 48.144.4-48.bg.1.62, 48.144.4-48.bg.1.63, 48.144.4-48.bg.1.64, 240.144.4-48.bg.1.1, 240.144.4-48.bg.1.2, 240.144.4-48.bg.1.3, 240.144.4-48.bg.1.4, 240.144.4-48.bg.1.5, 240.144.4-48.bg.1.6, 240.144.4-48.bg.1.7, 240.144.4-48.bg.1.8, 240.144.4-48.bg.1.9, 240.144.4-48.bg.1.10, 240.144.4-48.bg.1.11, 240.144.4-48.bg.1.12, 240.144.4-48.bg.1.13, 240.144.4-48.bg.1.14, 240.144.4-48.bg.1.15, 240.144.4-48.bg.1.16, 240.144.4-48.bg.1.17, 240.144.4-48.bg.1.18, 240.144.4-48.bg.1.19, 240.144.4-48.bg.1.20, 240.144.4-48.bg.1.21, 240.144.4-48.bg.1.22, 240.144.4-48.bg.1.23, 240.144.4-48.bg.1.24, 240.144.4-48.bg.1.25, 240.144.4-48.bg.1.26, 240.144.4-48.bg.1.27, 240.144.4-48.bg.1.28, 240.144.4-48.bg.1.29, 240.144.4-48.bg.1.30, 240.144.4-48.bg.1.31, 240.144.4-48.bg.1.32, 240.144.4-48.bg.1.33, 240.144.4-48.bg.1.34, 240.144.4-48.bg.1.35, 240.144.4-48.bg.1.36, 240.144.4-48.bg.1.37, 240.144.4-48.bg.1.38, 240.144.4-48.bg.1.39, 240.144.4-48.bg.1.40, 240.144.4-48.bg.1.41, 240.144.4-48.bg.1.42, 240.144.4-48.bg.1.43, 240.144.4-48.bg.1.44, 240.144.4-48.bg.1.45, 240.144.4-48.bg.1.46, 240.144.4-48.bg.1.47, 240.144.4-48.bg.1.48, 240.144.4-48.bg.1.49, 240.144.4-48.bg.1.50, 240.144.4-48.bg.1.51, 240.144.4-48.bg.1.52, 240.144.4-48.bg.1.53, 240.144.4-48.bg.1.54, 240.144.4-48.bg.1.55, 240.144.4-48.bg.1.56, 240.144.4-48.bg.1.57, 240.144.4-48.bg.1.58, 240.144.4-48.bg.1.59, 240.144.4-48.bg.1.60, 240.144.4-48.bg.1.61, 240.144.4-48.bg.1.62, 240.144.4-48.bg.1.63, 240.144.4-48.bg.1.64
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $16384$

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 $
$=$ $16 x y^{2} - x z^{2} - 3 w^{3}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 9 x^{5} + x z^{4} + 6 y^{3} z^{2} $
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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 between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{3^2}\cdot\frac{2277xyz^{7}w^{3}+9477xz^{2}w^{9}+65536y^{12}-256y^{2}z^{10}-5859yz^{5}w^{6}+16z^{12}-2187w^{12}}{w^{3}(xyz^{7}+81xz^{2}w^{6}+9yz^{5}w^{3}+81w^{9})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.36.2.cj.1 $24$ $2$ $2$ $2$ $0$ $1^{2}$
48.24.0.g.1 $48$ $3$ $3$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.144.7.eo.1 $48$ $2$ $2$ $7$ $0$ $1^{3}$
48.144.7.ep.1 $48$ $2$ $2$ $7$ $1$ $1^{3}$
48.144.7.es.1 $48$ $2$ $2$ $7$ $0$ $1^{3}$
48.144.7.et.1 $48$ $2$ $2$ $7$ $0$ $1^{3}$
48.144.7.fe.1 $48$ $2$ $2$ $7$ $1$ $1^{3}$
48.144.7.ff.1 $48$ $2$ $2$ $7$ $0$ $1^{3}$
48.144.7.fi.1 $48$ $2$ $2$ $7$ $1$ $1^{3}$
48.144.7.fj.1 $48$ $2$ $2$ $7$ $0$ $1^{3}$
48.144.8.hm.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hm.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hn.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hn.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ho.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ho.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hp.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hp.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hq.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hq.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hr.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hr.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hs.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hs.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ht.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ht.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hu.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hu.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hv.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hv.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hw.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hw.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hx.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hx.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hy.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hy.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hz.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.hz.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ia.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ia.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ib.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.ib.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.9.g.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.l.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.be.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.bh.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.cn.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.co.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.cr.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.cs.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.pc.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.pd.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.pg.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.ph.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.ps.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.pt.1 $48$ $2$ $2$ $9$ $0$ $1^{5}$
48.144.9.pw.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.px.1 $48$ $2$ $2$ $9$ $0$ $1^{5}$
144.216.16.bg.1 $144$ $3$ $3$ $16$ $?$ not computed
240.144.7.bdo.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bdp.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bds.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bdt.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bee.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bef.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bei.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.bej.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.8.xm.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xm.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xn.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xn.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xo.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xo.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xp.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xp.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xq.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xq.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xr.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xr.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xs.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xs.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xt.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xt.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xu.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xu.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xv.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xv.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xw.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xw.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xx.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xx.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xy.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xy.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xz.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.xz.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ya.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ya.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.yb.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.yb.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.9.bjk.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bjl.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bjo.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bjp.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bka.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bkb.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bke.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bkf.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.byy.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.byz.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzc.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzd.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzo.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzp.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzs.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bzt.1 $240$ $2$ $2$ $9$ $?$ not computed