$\GL_2(\Z/32\Z)$-generators: |
$\begin{bmatrix}1&13\\0&31\end{bmatrix}$, $\begin{bmatrix}5&2\\16&19\end{bmatrix}$, $\begin{bmatrix}5&3\\16&25\end{bmatrix}$, $\begin{bmatrix}19&14\\16&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
32.192.1-32.a.1.1, 32.192.1-32.a.1.2, 32.192.1-32.a.1.3, 32.192.1-32.a.1.4, 32.192.1-32.a.1.5, 32.192.1-32.a.1.6, 32.192.1-32.a.1.7, 32.192.1-32.a.1.8, 64.192.1-32.a.1.1, 64.192.1-32.a.1.2, 64.192.1-32.a.1.3, 64.192.1-32.a.1.4, 96.192.1-32.a.1.1, 96.192.1-32.a.1.2, 96.192.1-32.a.1.3, 96.192.1-32.a.1.4, 96.192.1-32.a.1.5, 96.192.1-32.a.1.6, 96.192.1-32.a.1.7, 96.192.1-32.a.1.8, 160.192.1-32.a.1.1, 160.192.1-32.a.1.2, 160.192.1-32.a.1.3, 160.192.1-32.a.1.4, 160.192.1-32.a.1.5, 160.192.1-32.a.1.6, 160.192.1-32.a.1.7, 160.192.1-32.a.1.8, 192.192.1-32.a.1.1, 192.192.1-32.a.1.2, 192.192.1-32.a.1.3, 192.192.1-32.a.1.4, 224.192.1-32.a.1.1, 224.192.1-32.a.1.2, 224.192.1-32.a.1.3, 224.192.1-32.a.1.4, 224.192.1-32.a.1.5, 224.192.1-32.a.1.6, 224.192.1-32.a.1.7, 224.192.1-32.a.1.8, 320.192.1-32.a.1.1, 320.192.1-32.a.1.2, 320.192.1-32.a.1.3, 320.192.1-32.a.1.4 |
Cyclic 32-isogeny field degree: |
$2$ |
Cyclic 32-torsion field degree: |
$16$ |
Full 32-torsion field degree: |
$4096$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + x $ |
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.
Maps to other modular curves
$j$-invariant map
of degree 96 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{732x^{2}y^{28}z^{2}-600270x^{2}y^{24}z^{6}-856619271x^{2}y^{20}z^{10}-63569039445x^{2}y^{16}z^{14}-1730871810984x^{2}y^{12}z^{18}+8645769167445x^{2}y^{8}z^{22}-3092376453141x^{2}y^{4}z^{26}+68719476735x^{2}z^{30}-8xy^{30}z-197586xy^{26}z^{5}-793368xy^{22}z^{9}+10317812503xy^{18}z^{13}+292057784784xy^{14}z^{17}-8568459754905xy^{10}z^{21}+8246337208300xy^{6}z^{25}-755914244097xy^{2}z^{29}-y^{32}+4488y^{28}z^{4}+21868132y^{24}z^{8}+1710112814y^{20}z^{12}+60129382116y^{16}z^{16}+3770981295824y^{12}z^{20}-5772436045246y^{8}z^{24}+687194767338y^{4}z^{28}-z^{32}}{z^{2}y^{8}(x^{2}y^{20}-13x^{2}y^{16}z^{4}+4x^{2}y^{12}z^{8}+39x^{2}y^{8}z^{12}+13x^{2}y^{4}z^{16}+x^{2}z^{20}-xy^{18}z^{3}-8xy^{14}z^{7}+29xy^{10}z^{11}+12xy^{6}z^{15}+xy^{2}z^{19}+6y^{20}z^{2}-26y^{16}z^{6}+24y^{12}z^{10}+50y^{8}z^{14}+14y^{4}z^{18}+z^{22})}$ |
Hi
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Cover information
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.