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Properties

Label 48.48.1.b.2
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.2

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&32\\32&33\end{bmatrix}$, $\begin{bmatrix}3&34\\32&31\end{bmatrix}$, $\begin{bmatrix}19&26\\24&1\end{bmatrix}$, $\begin{bmatrix}29&30\\16&31\end{bmatrix}$, $\begin{bmatrix}33&34\\16&43\end{bmatrix}$, $\begin{bmatrix}45&16\\16&43\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.b.2.1, 48.96.1-48.b.2.2, 48.96.1-48.b.2.3, 48.96.1-48.b.2.4, 48.96.1-48.b.2.5, 48.96.1-48.b.2.6, 48.96.1-48.b.2.7, 48.96.1-48.b.2.8, 48.96.1-48.b.2.9, 48.96.1-48.b.2.10, 48.96.1-48.b.2.11, 48.96.1-48.b.2.12, 48.96.1-48.b.2.13, 48.96.1-48.b.2.14, 48.96.1-48.b.2.15, 48.96.1-48.b.2.16, 48.96.1-48.b.2.17, 48.96.1-48.b.2.18, 48.96.1-48.b.2.19, 48.96.1-48.b.2.20, 48.96.1-48.b.2.21, 48.96.1-48.b.2.22, 48.96.1-48.b.2.23, 48.96.1-48.b.2.24, 240.96.1-48.b.2.1, 240.96.1-48.b.2.2, 240.96.1-48.b.2.3, 240.96.1-48.b.2.4, 240.96.1-48.b.2.5, 240.96.1-48.b.2.6, 240.96.1-48.b.2.7, 240.96.1-48.b.2.8, 240.96.1-48.b.2.9, 240.96.1-48.b.2.10, 240.96.1-48.b.2.11, 240.96.1-48.b.2.12, 240.96.1-48.b.2.13, 240.96.1-48.b.2.14, 240.96.1-48.b.2.15, 240.96.1-48.b.2.16, 240.96.1-48.b.2.17, 240.96.1-48.b.2.18, 240.96.1-48.b.2.19, 240.96.1-48.b.2.20, 240.96.1-48.b.2.21, 240.96.1-48.b.2.22, 240.96.1-48.b.2.23, 240.96.1-48.b.2.24
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

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} + 36x $
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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.

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

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^8}\cdot\frac{32400x^{2}y^{12}z^{2}-3083774976x^{2}y^{8}z^{6}+42316648611840x^{2}y^{4}z^{10}-288xy^{14}z+115893504xy^{10}z^{5}-3448023220224xy^{6}z^{9}+21936950640377856xy^{2}z^{13}+y^{16}-1772928y^{12}z^{4}+69657034752y^{8}z^{8}-609359740010496y^{4}z^{12}+4738381338321616896z^{16}}{z^{5}y^{4}(180x^{2}y^{4}z-6718464x^{2}z^{5}-xy^{6}+559872xy^{2}z^{4}-10368y^{4}z^{3})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.i.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.g.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.b.1 $48$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.96.1.g.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.g.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.i.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.i.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.p.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.p.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.r.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.r.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.3.bl.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.bu.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bu.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bx.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.cb.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.ck.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.ck.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cp.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.144.9.g.1 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.hr.1 $48$ $4$ $4$ $9$ $1$ $1^{8}$
240.96.1.s.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.s.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.u.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.u.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bh.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bh.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bj.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bj.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.ia.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.if.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.if.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ih.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.iy.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jh.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jh.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jm.1 $240$ $2$ $2$ $3$ $?$ not computed
240.240.17.d.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.n.2 $240$ $6$ $6$ $17$ $?$ not computed