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Properties

Label 48.24.1.b.1
Level $48$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $2^{2}\cdot4\cdot16$ Cusp orbits $1^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16A1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.24.1.2

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&11\\16&35\end{bmatrix}$, $\begin{bmatrix}17&11\\16&47\end{bmatrix}$, $\begin{bmatrix}23&41\\20&15\end{bmatrix}$, $\begin{bmatrix}29&23\\20&13\end{bmatrix}$, $\begin{bmatrix}31&23\\40&25\end{bmatrix}$, $\begin{bmatrix}31&32\\24&43\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.48.1-48.b.1.1, 48.48.1-48.b.1.2, 48.48.1-48.b.1.3, 48.48.1-48.b.1.4, 48.48.1-48.b.1.5, 48.48.1-48.b.1.6, 48.48.1-48.b.1.7, 48.48.1-48.b.1.8, 48.48.1-48.b.1.9, 48.48.1-48.b.1.10, 48.48.1-48.b.1.11, 48.48.1-48.b.1.12, 48.48.1-48.b.1.13, 48.48.1-48.b.1.14, 48.48.1-48.b.1.15, 48.48.1-48.b.1.16, 48.48.1-48.b.1.17, 48.48.1-48.b.1.18, 48.48.1-48.b.1.19, 48.48.1-48.b.1.20, 48.48.1-48.b.1.21, 48.48.1-48.b.1.22, 48.48.1-48.b.1.23, 48.48.1-48.b.1.24, 48.48.1-48.b.1.25, 48.48.1-48.b.1.26, 48.48.1-48.b.1.27, 48.48.1-48.b.1.28, 48.48.1-48.b.1.29, 48.48.1-48.b.1.30, 48.48.1-48.b.1.31, 48.48.1-48.b.1.32, 240.48.1-48.b.1.1, 240.48.1-48.b.1.2, 240.48.1-48.b.1.3, 240.48.1-48.b.1.4, 240.48.1-48.b.1.5, 240.48.1-48.b.1.6, 240.48.1-48.b.1.7, 240.48.1-48.b.1.8, 240.48.1-48.b.1.9, 240.48.1-48.b.1.10, 240.48.1-48.b.1.11, 240.48.1-48.b.1.12, 240.48.1-48.b.1.13, 240.48.1-48.b.1.14, 240.48.1-48.b.1.15, 240.48.1-48.b.1.16, 240.48.1-48.b.1.17, 240.48.1-48.b.1.18, 240.48.1-48.b.1.19, 240.48.1-48.b.1.20, 240.48.1-48.b.1.21, 240.48.1-48.b.1.22, 240.48.1-48.b.1.23, 240.48.1-48.b.1.24, 240.48.1-48.b.1.25, 240.48.1-48.b.1.26, 240.48.1-48.b.1.27, 240.48.1-48.b.1.28, 240.48.1-48.b.1.29, 240.48.1-48.b.1.30, 240.48.1-48.b.1.31, 240.48.1-48.b.1.32
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $49152$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 9x $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(0:0:1)$, $(3:0:1)$, $(-3:0:1)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^8}\cdot\frac{62208x^{2}y^{4}z^{2}+36864xy^{6}z+314928xy^{2}z^{5}+4096y^{8}+531441z^{8}}{z^{5}y^{2}x}$

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
$X_0(8)$ $8$ $2$ $2$ $0$ $0$ 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
48.48.1.b.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.f.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.h.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.j.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bw.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bw.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bx.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bx.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.by.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.by.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bz.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bz.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ca.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ca.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cb.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cb.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cc.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cc.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cd.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cd.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ce.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ch.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ci.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cl.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.72.5.f.1 $48$ $3$ $3$ $5$ $1$ $1^{4}$
48.96.5.oq.1 $48$ $4$ $4$ $5$ $1$ $1^{4}$
240.48.1.ci.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.cj.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.cm.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.cn.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.de.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.de.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.df.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.df.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dg.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dg.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dh.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dh.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.di.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.di.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dj.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dj.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dk.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dk.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dl.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.dl.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.ee.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.ef.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.ei.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.48.1.ej.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.120.9.b.1 $240$ $5$ $5$ $9$ $?$ not computed
240.144.9.mj.1 $240$ $6$ $6$ $9$ $?$ not computed
240.240.17.fr.1 $240$ $10$ $10$ $17$ $?$ not computed