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Properties

Label 32.96.1.b.2
Level $32$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $32$ $\SL_2$-level: $32$ Newform level: $64$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $1^{8}\cdot2^{4}\cdot8^{2}\cdot32^{2}$ Cusp orbits $2^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 32E1
Rouse and Zureick-Brown (RZB) label: X489
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.96.1.39

Level structure

$\GL_2(\Z/32\Z)$-generators: $\begin{bmatrix}3&12\\16&7\end{bmatrix}$, $\begin{bmatrix}5&26\\16&9\end{bmatrix}$, $\begin{bmatrix}15&14\\0&25\end{bmatrix}$, $\begin{bmatrix}23&9\\0&23\end{bmatrix}$, $\begin{bmatrix}31&31\\0&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 32.192.1-32.b.2.1, 32.192.1-32.b.2.2, 32.192.1-32.b.2.3, 32.192.1-32.b.2.4, 32.192.1-32.b.2.5, 32.192.1-32.b.2.6, 32.192.1-32.b.2.7, 32.192.1-32.b.2.8, 32.192.1-32.b.2.9, 32.192.1-32.b.2.10, 32.192.1-32.b.2.11, 32.192.1-32.b.2.12, 32.192.1-32.b.2.13, 32.192.1-32.b.2.14, 32.192.1-32.b.2.15, 32.192.1-32.b.2.16, 96.192.1-32.b.2.1, 96.192.1-32.b.2.2, 96.192.1-32.b.2.3, 96.192.1-32.b.2.4, 96.192.1-32.b.2.5, 96.192.1-32.b.2.6, 96.192.1-32.b.2.7, 96.192.1-32.b.2.8, 96.192.1-32.b.2.9, 96.192.1-32.b.2.10, 96.192.1-32.b.2.11, 96.192.1-32.b.2.12, 96.192.1-32.b.2.13, 96.192.1-32.b.2.14, 96.192.1-32.b.2.15, 96.192.1-32.b.2.16, 160.192.1-32.b.2.1, 160.192.1-32.b.2.2, 160.192.1-32.b.2.3, 160.192.1-32.b.2.4, 160.192.1-32.b.2.5, 160.192.1-32.b.2.6, 160.192.1-32.b.2.7, 160.192.1-32.b.2.8, 160.192.1-32.b.2.9, 160.192.1-32.b.2.10, 160.192.1-32.b.2.11, 160.192.1-32.b.2.12, 160.192.1-32.b.2.13, 160.192.1-32.b.2.14, 160.192.1-32.b.2.15, 160.192.1-32.b.2.16, 224.192.1-32.b.2.1, 224.192.1-32.b.2.2, 224.192.1-32.b.2.3, 224.192.1-32.b.2.4, 224.192.1-32.b.2.5, 224.192.1-32.b.2.6, 224.192.1-32.b.2.7, 224.192.1-32.b.2.8, 224.192.1-32.b.2.9, 224.192.1-32.b.2.10, 224.192.1-32.b.2.11, 224.192.1-32.b.2.12, 224.192.1-32.b.2.13, 224.192.1-32.b.2.14, 224.192.1-32.b.2.15, 224.192.1-32.b.2.16
Cyclic 32-isogeny field degree: $2$
Cyclic 32-torsion field degree: $16$
Full 32-torsion field degree: $4096$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} - x y + z^{2} $
$=$ $2 x^{2} - x y + 2 y^{2} - 7 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{2} y^{2} + 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{4}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4}\cdot\frac{94371840xyz^{20}w^{2}-2173501440xyz^{16}w^{6}+638484480xyz^{12}w^{10}-34606080xyz^{8}w^{14}+655200xyz^{4}w^{18}-4095xyw^{22}+4194304z^{24}-94371840z^{22}w^{2}-746323968z^{20}w^{4}+2173501440z^{18}w^{6}+1888272384z^{16}w^{8}-638484480z^{14}w^{10}-264220672z^{12}w^{12}+34606080z^{10}w^{14}+11010624z^{8}w^{16}-655200z^{6}w^{18}-180228z^{4}w^{20}+4095z^{2}w^{22}+1024w^{24}}{w^{2}z^{8}(8192xyz^{12}-2304xyz^{8}w^{4}+96xyz^{4}w^{8}-xyw^{12}-8192z^{14}+5120z^{12}w^{2}+2304z^{10}w^{4}-320z^{8}w^{6}-96z^{6}w^{8}+4z^{4}w^{10}+z^{2}w^{12})}$

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
16.48.0.v.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
32.48.0.e.1 $32$ $2$ $2$ $0$ $0$ full Jacobian
32.48.1.a.1 $32$ $2$ $2$ $1$ $0$ 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
32.192.5.e.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.f.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.v.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.z.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.bh.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bi.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bj.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bk.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bp.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bq.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.br.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bs.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
96.192.5.fk.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.fo.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ga.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ge.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.hv.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.hw.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.hx.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.hy.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.id.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ie.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.if.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ig.2 $96$ $2$ $2$ $5$ $?$ not computed
96.288.17.fd.2 $96$ $3$ $3$ $17$ $?$ not computed
96.384.17.qb.2 $96$ $4$ $4$ $17$ $?$ not computed
160.192.5.iu.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.iy.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.jk.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.jo.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nb.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nc.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nd.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ne.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nj.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nk.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nl.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.nm.2 $160$ $2$ $2$ $5$ $?$ not computed
224.192.5.fk.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.fo.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.ga.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.ge.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.hv.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.hw.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.hx.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.hy.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.id.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.ie.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.if.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.ig.2 $224$ $2$ $2$ $5$ $?$ not computed