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Properties

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

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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: $16$ (of which $2$ are rational) Cusp widths $1^{8}\cdot2^{4}\cdot8^{2}\cdot32^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
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: 32E1
Rouse and Zureick-Brown (RZB) label: X491
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.96.1.27

Level structure

$\GL_2(\Z/32\Z)$-generators: $\begin{bmatrix}13&1\\0&11\end{bmatrix}$, $\begin{bmatrix}17&2\\0&21\end{bmatrix}$, $\begin{bmatrix}19&3\\16&19\end{bmatrix}$, $\begin{bmatrix}25&6\\16&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 32.192.1-32.d.1.1, 32.192.1-32.d.1.2, 32.192.1-32.d.1.3, 32.192.1-32.d.1.4, 32.192.1-32.d.1.5, 32.192.1-32.d.1.6, 32.192.1-32.d.1.7, 32.192.1-32.d.1.8, 64.192.1-32.d.1.1, 64.192.1-32.d.1.2, 64.192.1-32.d.1.3, 64.192.1-32.d.1.4, 96.192.1-32.d.1.1, 96.192.1-32.d.1.2, 96.192.1-32.d.1.3, 96.192.1-32.d.1.4, 96.192.1-32.d.1.5, 96.192.1-32.d.1.6, 96.192.1-32.d.1.7, 96.192.1-32.d.1.8, 160.192.1-32.d.1.1, 160.192.1-32.d.1.2, 160.192.1-32.d.1.3, 160.192.1-32.d.1.4, 160.192.1-32.d.1.5, 160.192.1-32.d.1.6, 160.192.1-32.d.1.7, 160.192.1-32.d.1.8, 192.192.1-32.d.1.1, 192.192.1-32.d.1.2, 192.192.1-32.d.1.3, 192.192.1-32.d.1.4, 224.192.1-32.d.1.1, 224.192.1-32.d.1.2, 224.192.1-32.d.1.3, 224.192.1-32.d.1.4, 224.192.1-32.d.1.5, 224.192.1-32.d.1.6, 224.192.1-32.d.1.7, 224.192.1-32.d.1.8, 320.192.1-32.d.1.1, 320.192.1-32.d.1.2, 320.192.1-32.d.1.3, 320.192.1-32.d.1.4
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

Weierstrass model Weierstrass model

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{12x^{2}y^{28}z^{2}-1230x^{2}y^{24}z^{6}-14391x^{2}y^{20}z^{10}-47925x^{2}y^{16}z^{14}+69576x^{2}y^{12}z^{18}+503685x^{2}y^{8}z^{22}-184341x^{2}y^{4}z^{26}+4095x^{2}z^{30}-8xy^{30}z-306xy^{26}z^{5}-1368xy^{22}z^{9}-26537xy^{18}z^{13}-212976xy^{14}z^{17}-430185xy^{10}z^{21}+491500xy^{6}z^{25}-45057xy^{2}z^{29}-y^{32}+168y^{28}z^{4}+3172y^{24}z^{8}+24974y^{20}z^{12}+119316y^{16}z^{16}+155504y^{12}z^{20}-344206y^{8}z^{24}+40938y^{4}z^{28}-z^{32}}{z^{10}y^{8}(x^{2}y^{12}+79x^{2}y^{8}z^{4}-240x^{2}y^{4}z^{8}+16x^{2}z^{12}-13xy^{10}z^{3}+320xy^{6}z^{7}-112xy^{2}z^{11}-2y^{12}z^{2}-160y^{8}z^{6}+96y^{4}z^{10})}$

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.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
32.48.0.e.1 $32$ $2$ $2$ $0$ $0$ full Jacobian
32.48.1.a.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.3 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.g.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.x.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.z.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
64.192.5.d.1 $64$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
64.192.5.d.2 $64$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
64.192.9.r.1 $64$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{3}$
64.192.9.r.2 $64$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{3}$
96.192.5.fm.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.fq.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.gc.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.gg.1 $96$ $2$ $2$ $5$ $?$ not computed
96.288.17.ff.2 $96$ $3$ $3$ $17$ $?$ not computed
96.384.17.qd.1 $96$ $4$ $4$ $17$ $?$ not computed
160.192.5.iw.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ja.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.jm.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.jq.1 $160$ $2$ $2$ $5$ $?$ not computed
192.192.5.d.2 $192$ $2$ $2$ $5$ $?$ not computed
192.192.5.d.3 $192$ $2$ $2$ $5$ $?$ not computed
192.192.9.cf.1 $192$ $2$ $2$ $9$ $?$ not computed
192.192.9.cf.2 $192$ $2$ $2$ $9$ $?$ not computed
224.192.5.fm.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.fq.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.gc.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.gg.1 $224$ $2$ $2$ $5$ $?$ not computed
320.192.5.h.3 $320$ $2$ $2$ $5$ $?$ not computed
320.192.5.h.4 $320$ $2$ $2$ $5$ $?$ not computed
320.192.9.cf.1 $320$ $2$ $2$ $9$ $?$ not computed
320.192.9.cf.2 $320$ $2$ $2$ $9$ $?$ not computed