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Properties

Label 48.48.1.b.1
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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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 $4$ are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $1^{4}\cdot4$
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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.3

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&26\\16&45\end{bmatrix}$, $\begin{bmatrix}11&12\\32&31\end{bmatrix}$, $\begin{bmatrix}15&22\\8&39\end{bmatrix}$, $\begin{bmatrix}15&40\\40&15\end{bmatrix}$, $\begin{bmatrix}25&30\\8&11\end{bmatrix}$, $\begin{bmatrix}27&46\\40&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.b.1.1, 48.96.1-48.b.1.2, 48.96.1-48.b.1.3, 48.96.1-48.b.1.4, 48.96.1-48.b.1.5, 48.96.1-48.b.1.6, 48.96.1-48.b.1.7, 48.96.1-48.b.1.8, 48.96.1-48.b.1.9, 48.96.1-48.b.1.10, 48.96.1-48.b.1.11, 48.96.1-48.b.1.12, 48.96.1-48.b.1.13, 48.96.1-48.b.1.14, 48.96.1-48.b.1.15, 48.96.1-48.b.1.16, 48.96.1-48.b.1.17, 48.96.1-48.b.1.18, 48.96.1-48.b.1.19, 48.96.1-48.b.1.20, 48.96.1-48.b.1.21, 48.96.1-48.b.1.22, 48.96.1-48.b.1.23, 48.96.1-48.b.1.24, 240.96.1-48.b.1.1, 240.96.1-48.b.1.2, 240.96.1-48.b.1.3, 240.96.1-48.b.1.4, 240.96.1-48.b.1.5, 240.96.1-48.b.1.6, 240.96.1-48.b.1.7, 240.96.1-48.b.1.8, 240.96.1-48.b.1.9, 240.96.1-48.b.1.10, 240.96.1-48.b.1.11, 240.96.1-48.b.1.12, 240.96.1-48.b.1.13, 240.96.1-48.b.1.14, 240.96.1-48.b.1.15, 240.96.1-48.b.1.16, 240.96.1-48.b.1.17, 240.96.1-48.b.1.18, 240.96.1-48.b.1.19, 240.96.1-48.b.1.20, 240.96.1-48.b.1.21, 240.96.1-48.b.1.22, 240.96.1-48.b.1.23, 240.96.1-48.b.1.24
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} - 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
$(0:1:0)$, $(-3:0:1)$, $(0:0:1)$, $(3:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3^8}\cdot\frac{2025x^{2}y^{12}z^{2}+3011499x^{2}y^{8}z^{6}+645700815x^{2}y^{4}z^{10}+72xy^{14}z+452709xy^{10}z^{5}+210450636xy^{6}z^{9}+20920706406xy^{2}z^{13}+y^{16}+27702y^{12}z^{4}+17006112y^{8}z^{8}+2324522934y^{4}z^{12}+282429536481z^{16}}{z^{5}y^{4}(45x^{2}y^{4}z+26244x^{2}z^{5}+xy^{6}+8748xy^{2}z^{4}+648y^{4}z^{3})}$

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.i.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.d.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.h.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.h.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.q.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.q.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.3.bl.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.bs.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bs.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bv.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bv.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bx.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.cb.2 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.cj.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cj.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cl.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cl.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cp.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.144.9.g.2 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.hr.2 $48$ $4$ $4$ $9$ $1$ $1^{8}$
240.96.1.t.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.t.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bi.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.bi.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.ia.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ie.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ie.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ig.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ig.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ih.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.iy.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jg.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jg.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ji.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ji.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.jm.2 $240$ $2$ $2$ $3$ $?$ not computed
240.240.17.d.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.n.1 $240$ $6$ $6$ $17$ $?$ not computed