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Properties

Label 32.48.1.b.2
Level $32$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $32$ $\SL_2$-level: $32$ Newform level: $32$
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 $1^{4}\cdot2^{2}\cdot8\cdot32$ Cusp orbits $1^{4}\cdot2^{2}$
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: 32A1
Rouse and Zureick-Brown (RZB) label: X353
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.48.1.4

Level structure

$\GL_2(\Z/32\Z)$-generators: $\begin{bmatrix}5&13\\16&13\end{bmatrix}$, $\begin{bmatrix}7&9\\16&17\end{bmatrix}$, $\begin{bmatrix}7&31\\0&27\end{bmatrix}$, $\begin{bmatrix}21&27\\0&9\end{bmatrix}$, $\begin{bmatrix}31&10\\0&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 32.96.1-32.b.2.1, 32.96.1-32.b.2.2, 32.96.1-32.b.2.3, 32.96.1-32.b.2.4, 32.96.1-32.b.2.5, 32.96.1-32.b.2.6, 32.96.1-32.b.2.7, 32.96.1-32.b.2.8, 32.96.1-32.b.2.9, 32.96.1-32.b.2.10, 32.96.1-32.b.2.11, 32.96.1-32.b.2.12, 32.96.1-32.b.2.13, 32.96.1-32.b.2.14, 32.96.1-32.b.2.15, 32.96.1-32.b.2.16, 96.96.1-32.b.2.1, 96.96.1-32.b.2.2, 96.96.1-32.b.2.3, 96.96.1-32.b.2.4, 96.96.1-32.b.2.5, 96.96.1-32.b.2.6, 96.96.1-32.b.2.7, 96.96.1-32.b.2.8, 96.96.1-32.b.2.9, 96.96.1-32.b.2.10, 96.96.1-32.b.2.11, 96.96.1-32.b.2.12, 96.96.1-32.b.2.13, 96.96.1-32.b.2.14, 96.96.1-32.b.2.15, 96.96.1-32.b.2.16, 160.96.1-32.b.2.1, 160.96.1-32.b.2.2, 160.96.1-32.b.2.3, 160.96.1-32.b.2.4, 160.96.1-32.b.2.5, 160.96.1-32.b.2.6, 160.96.1-32.b.2.7, 160.96.1-32.b.2.8, 160.96.1-32.b.2.9, 160.96.1-32.b.2.10, 160.96.1-32.b.2.11, 160.96.1-32.b.2.12, 160.96.1-32.b.2.13, 160.96.1-32.b.2.14, 160.96.1-32.b.2.15, 160.96.1-32.b.2.16, 224.96.1-32.b.2.1, 224.96.1-32.b.2.2, 224.96.1-32.b.2.3, 224.96.1-32.b.2.4, 224.96.1-32.b.2.5, 224.96.1-32.b.2.6, 224.96.1-32.b.2.7, 224.96.1-32.b.2.8, 224.96.1-32.b.2.9, 224.96.1-32.b.2.10, 224.96.1-32.b.2.11, 224.96.1-32.b.2.12, 224.96.1-32.b.2.13, 224.96.1-32.b.2.14, 224.96.1-32.b.2.15, 224.96.1-32.b.2.16
Cyclic 32-isogeny field degree: $2$
Cyclic 32-torsion field degree: $32$
Full 32-torsion field degree: $8192$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x $
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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
$(0:1:0)$, $(1:0:1)$, $(0:0:1)$, $(-1:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{102400x^{2}y^{12}z^{2}+151296x^{2}y^{8}z^{6}+3120x^{2}y^{4}z^{10}+32768xy^{14}z+271104xy^{10}z^{5}+41984xy^{6}z^{9}+96xy^{2}z^{13}+4096y^{16}+155648y^{12}z^{4}+38912y^{8}z^{8}+96y^{4}z^{12}+z^{16}}{z^{10}y^{2}(x^{2}y^{2}+2xz^{3}+2y^{2}z^{2})}$

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
$X_0(16)$ $16$ $2$ $2$ $0$ $0$ 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
32.96.1.e.1 $32$ $2$ $2$ $1$ $0$ dimension zero
32.96.1.e.2 $32$ $2$ $2$ $1$ $0$ dimension zero
32.96.1.h.1 $32$ $2$ $2$ $1$ $0$ dimension zero
32.96.1.h.2 $32$ $2$ $2$ $1$ $0$ dimension zero
32.96.3.d.2 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.g.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.j.2 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.m.1 $32$ $2$ $2$ $3$ $0$ $1^{2}$
32.96.3.w.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.w.2 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.x.1 $32$ $2$ $2$ $3$ $0$ $2$
32.96.3.x.2 $32$ $2$ $2$ $3$ $0$ $2$
96.96.1.m.1 $96$ $2$ $2$ $1$ $?$ dimension zero
96.96.1.m.2 $96$ $2$ $2$ $1$ $?$ dimension zero
96.96.1.p.1 $96$ $2$ $2$ $1$ $?$ dimension zero
96.96.1.p.2 $96$ $2$ $2$ $1$ $?$ dimension zero
96.96.3.bn.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.bo.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.br.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.bs.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.cg.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.cg.2 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.ch.1 $96$ $2$ $2$ $3$ $?$ not computed
96.96.3.ch.2 $96$ $2$ $2$ $3$ $?$ not computed
96.144.9.h.2 $96$ $3$ $3$ $9$ $?$ not computed
96.192.9.mo.2 $96$ $4$ $4$ $9$ $?$ not computed
160.96.1.m.1 $160$ $2$ $2$ $1$ $?$ dimension zero
160.96.1.m.2 $160$ $2$ $2$ $1$ $?$ dimension zero
160.96.1.p.1 $160$ $2$ $2$ $1$ $?$ dimension zero
160.96.1.p.2 $160$ $2$ $2$ $1$ $?$ dimension zero
160.96.3.bz.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.ca.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.cd.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.ce.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.cs.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.cs.2 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.ct.1 $160$ $2$ $2$ $3$ $?$ not computed
160.96.3.ct.2 $160$ $2$ $2$ $3$ $?$ not computed
160.240.17.d.2 $160$ $5$ $5$ $17$ $?$ not computed
160.288.17.f.2 $160$ $6$ $6$ $17$ $?$ not computed
224.96.1.m.1 $224$ $2$ $2$ $1$ $?$ dimension zero
224.96.1.m.2 $224$ $2$ $2$ $1$ $?$ dimension zero
224.96.1.p.1 $224$ $2$ $2$ $1$ $?$ dimension zero
224.96.1.p.2 $224$ $2$ $2$ $1$ $?$ dimension zero
224.96.3.bn.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.bo.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.br.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.bs.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.cg.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.cg.2 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.ch.1 $224$ $2$ $2$ $3$ $?$ not computed
224.96.3.ch.2 $224$ $2$ $2$ $3$ $?$ not computed