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Properties

Label 48.144.8.hp.2
Level $48$
Index $144$
Genus $8$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $144$
Index: $144$ $\PSL_2$-index:$144$
Genus: $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $2$ are rational) Cusp widths $3^{4}\cdot6^{2}\cdot12^{2}\cdot48^{2}$ Cusp orbits $1^{2}\cdot2^{2}\cdot4$
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: 48E8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.144.8.493

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}13&43\\8&31\end{bmatrix}$, $\begin{bmatrix}19&21\\0&1\end{bmatrix}$, $\begin{bmatrix}19&34\\40&35\end{bmatrix}$, $\begin{bmatrix}21&31\\8&3\end{bmatrix}$, $\begin{bmatrix}35&45\\0&37\end{bmatrix}$, $\begin{bmatrix}41&45\\12&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.288.8-48.hp.2.1, 48.288.8-48.hp.2.2, 48.288.8-48.hp.2.3, 48.288.8-48.hp.2.4, 48.288.8-48.hp.2.5, 48.288.8-48.hp.2.6, 48.288.8-48.hp.2.7, 48.288.8-48.hp.2.8, 48.288.8-48.hp.2.9, 48.288.8-48.hp.2.10, 48.288.8-48.hp.2.11, 48.288.8-48.hp.2.12, 48.288.8-48.hp.2.13, 48.288.8-48.hp.2.14, 48.288.8-48.hp.2.15, 48.288.8-48.hp.2.16, 48.288.8-48.hp.2.17, 48.288.8-48.hp.2.18, 48.288.8-48.hp.2.19, 48.288.8-48.hp.2.20, 48.288.8-48.hp.2.21, 48.288.8-48.hp.2.22, 48.288.8-48.hp.2.23, 48.288.8-48.hp.2.24, 48.288.8-48.hp.2.25, 48.288.8-48.hp.2.26, 48.288.8-48.hp.2.27, 48.288.8-48.hp.2.28, 48.288.8-48.hp.2.29, 48.288.8-48.hp.2.30, 48.288.8-48.hp.2.31, 48.288.8-48.hp.2.32, 240.288.8-48.hp.2.1, 240.288.8-48.hp.2.2, 240.288.8-48.hp.2.3, 240.288.8-48.hp.2.4, 240.288.8-48.hp.2.5, 240.288.8-48.hp.2.6, 240.288.8-48.hp.2.7, 240.288.8-48.hp.2.8, 240.288.8-48.hp.2.9, 240.288.8-48.hp.2.10, 240.288.8-48.hp.2.11, 240.288.8-48.hp.2.12, 240.288.8-48.hp.2.13, 240.288.8-48.hp.2.14, 240.288.8-48.hp.2.15, 240.288.8-48.hp.2.16, 240.288.8-48.hp.2.17, 240.288.8-48.hp.2.18, 240.288.8-48.hp.2.19, 240.288.8-48.hp.2.20, 240.288.8-48.hp.2.21, 240.288.8-48.hp.2.22, 240.288.8-48.hp.2.23, 240.288.8-48.hp.2.24, 240.288.8-48.hp.2.25, 240.288.8-48.hp.2.26, 240.288.8-48.hp.2.27, 240.288.8-48.hp.2.28, 240.288.8-48.hp.2.29, 240.288.8-48.hp.2.30, 240.288.8-48.hp.2.31, 240.288.8-48.hp.2.32
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $8192$

Jacobian

Conductor: $2^{22}\cdot3^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{2}$
Newforms: 36.2.a.a$^{3}$, 72.2.d.a$^{2}$, 144.2.a.a

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations

$ 0 $ $=$ $ y r - t v $
$=$ $x w - y z$
$=$ $x r - z t + w t$
$=$ $x r - y r - t v + u v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 27 x^{6} y^{3} + 63 x^{4} y^{3} z^{2} + 9 x^{4} z^{5} + 21 x^{2} y^{3} z^{4} - 3 x^{2} z^{7} + y^{3} z^{6} $
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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
$(0:0:1:0:0:0:0:0)$, $(0:0:1:1:0:0:0:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle t$
$\displaystyle Y$ $=$ $\displaystyle v$
$\displaystyle Z$ $=$ $\displaystyle r$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.72.4.gl.1 :

$\displaystyle X$ $=$ $\displaystyle -y$
$\displaystyle Y$ $=$ $\displaystyle -x+y$
$\displaystyle Z$ $=$ $\displaystyle -t$
$\displaystyle W$ $=$ $\displaystyle -u$

Equation of the image curve:

$0$ $=$ $ XZ-YZ-XW $
$=$ $ X^{3}+7X^{2}Y+7XY^{2}+Y^{3}+Z^{2}W-ZW^{2} $

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.72.4.gl.1 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
48.48.0.bf.1 $48$ $3$ $3$ $0$ $0$ full Jacobian
48.72.4.z.2 $48$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
48.72.4.bg.1 $48$ $2$ $2$ $4$ $0$ $2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.288.15.sl.1 $48$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
48.288.15.st.1 $48$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
48.288.15.tr.2 $48$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
48.288.15.tz.1 $48$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
48.288.15.ux.2 $48$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
48.288.15.vh.2 $48$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
48.288.15.wl.2 $48$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
48.288.15.wt.1 $48$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
48.288.17.ea.1 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.gn.1 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.mk.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.ov.1 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.bad.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.bar.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.bbp.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.bbr.1 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.cpz.1 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.cqh.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.crf.1 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.crn.1 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.csp.1 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.csx.1 $48$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
48.288.17.ctw.1 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.cuf.2 $48$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
240.288.15.fwd.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.fwt.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.fyp.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.fzf.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.gbb.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.gbr.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.gdn.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.15.ged.1 $240$ $2$ $2$ $15$ $?$ not computed
240.288.17.qpj.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qpz.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qrv.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qsl.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.quh.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qux.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qwt.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.qxj.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ufn.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ugd.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.uhz.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.uip.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ukl.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ulb.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.umx.1 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.unn.2 $240$ $2$ $2$ $17$ $?$ not computed