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Properties

Label 32.96.1.h.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: $32$
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: X497
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.96.1.37

Level structure

$\GL_2(\Z/32\Z)$-generators: $\begin{bmatrix}3&8\\16&15\end{bmatrix}$, $\begin{bmatrix}5&22\\16&9\end{bmatrix}$, $\begin{bmatrix}9&11\\0&11\end{bmatrix}$, $\begin{bmatrix}25&20\\0&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 32.192.1-32.h.2.1, 32.192.1-32.h.2.2, 32.192.1-32.h.2.3, 32.192.1-32.h.2.4, 32.192.1-32.h.2.5, 32.192.1-32.h.2.6, 32.192.1-32.h.2.7, 32.192.1-32.h.2.8, 96.192.1-32.h.2.1, 96.192.1-32.h.2.2, 96.192.1-32.h.2.3, 96.192.1-32.h.2.4, 96.192.1-32.h.2.5, 96.192.1-32.h.2.6, 96.192.1-32.h.2.7, 96.192.1-32.h.2.8, 160.192.1-32.h.2.1, 160.192.1-32.h.2.2, 160.192.1-32.h.2.3, 160.192.1-32.h.2.4, 160.192.1-32.h.2.5, 160.192.1-32.h.2.6, 160.192.1-32.h.2.7, 160.192.1-32.h.2.8, 224.192.1-32.h.2.1, 224.192.1-32.h.2.2, 224.192.1-32.h.2.3, 224.192.1-32.h.2.4, 224.192.1-32.h.2.5, 224.192.1-32.h.2.6, 224.192.1-32.h.2.7, 224.192.1-32.h.2.8
Cyclic 32-isogeny field degree: $2$
Cyclic 32-torsion field degree: $16$
Full 32-torsion field degree: $4096$

Jacobian

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

Models

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

$ 0 $ $=$ $ x^{2} - x y + z^{2} $
$=$ $5 x^{2} + 5 x y - 2 y^{2} - z^{2} + w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{5}{2^4}\cdot\frac{37748736y^{2}z^{20}w^{2}+869400576y^{2}z^{16}w^{6}+255393792y^{2}z^{12}w^{10}+13842432y^{2}z^{8}w^{14}+262080y^{2}z^{4}w^{18}+1638y^{2}w^{22}+4194304z^{24}-75497472z^{22}w^{2}+727449600z^{20}w^{4}-1738801152z^{18}w^{6}+1453572096z^{16}w^{8}-510787584z^{14}w^{10}+136523776z^{12}w^{12}-27684864z^{10}w^{14}+4089408z^{8}w^{16}-524160z^{6}w^{18}+49188z^{4}w^{20}-3276z^{2}w^{22}+205w^{24}}{w^{2}z^{8}(16384y^{2}z^{12}+4608y^{2}z^{8}w^{4}+192y^{2}z^{4}w^{8}+2y^{2}w^{12}-32768z^{14}-33792z^{12}w^{2}-9216z^{10}w^{4}-3904z^{8}w^{6}-384z^{6}w^{8}-116z^{4}w^{10}-4z^{2}w^{12}-w^{14})}$

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.x.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
32.48.0.e.1 $32$ $2$ $2$ $0$ $0$ full Jacobian
32.48.1.b.2 $32$ $2$ $2$ $1$ $0$ 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.k.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.x.4 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.bb.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
96.192.5.fu.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.fy.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.gk.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.go.1 $96$ $2$ $2$ $5$ $?$ not computed
96.288.17.gh.2 $96$ $3$ $3$ $17$ $?$ not computed
96.384.17.qp.2 $96$ $4$ $4$ $17$ $?$ not computed
160.192.5.je.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ji.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ju.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.jy.2 $160$ $2$ $2$ $5$ $?$ not computed
224.192.5.fu.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.fy.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.gk.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.go.2 $224$ $2$ $2$ $5$ $?$ not computed