$\GL_2(\Z/32\Z)$-generators: |
$\begin{bmatrix}7&4\\0&23\end{bmatrix}$, $\begin{bmatrix}7&5\\0&15\end{bmatrix}$, $\begin{bmatrix}7&6\\0&13\end{bmatrix}$, $\begin{bmatrix}9&8\\0&27\end{bmatrix}$, $\begin{bmatrix}25&27\\0&31\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
32.192.1-32.f.1.1, 32.192.1-32.f.1.2, 32.192.1-32.f.1.3, 32.192.1-32.f.1.4, 32.192.1-32.f.1.5, 32.192.1-32.f.1.6, 32.192.1-32.f.1.7, 32.192.1-32.f.1.8, 32.192.1-32.f.1.9, 32.192.1-32.f.1.10, 32.192.1-32.f.1.11, 32.192.1-32.f.1.12, 32.192.1-32.f.1.13, 32.192.1-32.f.1.14, 32.192.1-32.f.1.15, 32.192.1-32.f.1.16, 64.192.1-32.f.1.1, 64.192.1-32.f.1.2, 64.192.1-32.f.1.3, 64.192.1-32.f.1.4, 64.192.1-32.f.1.5, 64.192.1-32.f.1.6, 64.192.1-32.f.1.7, 64.192.1-32.f.1.8, 96.192.1-32.f.1.1, 96.192.1-32.f.1.2, 96.192.1-32.f.1.3, 96.192.1-32.f.1.4, 96.192.1-32.f.1.5, 96.192.1-32.f.1.6, 96.192.1-32.f.1.7, 96.192.1-32.f.1.8, 96.192.1-32.f.1.9, 96.192.1-32.f.1.10, 96.192.1-32.f.1.11, 96.192.1-32.f.1.12, 96.192.1-32.f.1.13, 96.192.1-32.f.1.14, 96.192.1-32.f.1.15, 96.192.1-32.f.1.16, 160.192.1-32.f.1.1, 160.192.1-32.f.1.2, 160.192.1-32.f.1.3, 160.192.1-32.f.1.4, 160.192.1-32.f.1.5, 160.192.1-32.f.1.6, 160.192.1-32.f.1.7, 160.192.1-32.f.1.8, 160.192.1-32.f.1.9, 160.192.1-32.f.1.10, 160.192.1-32.f.1.11, 160.192.1-32.f.1.12, 160.192.1-32.f.1.13, 160.192.1-32.f.1.14, 160.192.1-32.f.1.15, 160.192.1-32.f.1.16, 192.192.1-32.f.1.1, 192.192.1-32.f.1.2, 192.192.1-32.f.1.3, 192.192.1-32.f.1.4, 192.192.1-32.f.1.5, 192.192.1-32.f.1.6, 192.192.1-32.f.1.7, 192.192.1-32.f.1.8, 224.192.1-32.f.1.1, 224.192.1-32.f.1.2, 224.192.1-32.f.1.3, 224.192.1-32.f.1.4, 224.192.1-32.f.1.5, 224.192.1-32.f.1.6, 224.192.1-32.f.1.7, 224.192.1-32.f.1.8, 224.192.1-32.f.1.9, 224.192.1-32.f.1.10, 224.192.1-32.f.1.11, 224.192.1-32.f.1.12, 224.192.1-32.f.1.13, 224.192.1-32.f.1.14, 224.192.1-32.f.1.15, 224.192.1-32.f.1.16, 320.192.1-32.f.1.1, 320.192.1-32.f.1.2, 320.192.1-32.f.1.3, 320.192.1-32.f.1.4, 320.192.1-32.f.1.5, 320.192.1-32.f.1.6, 320.192.1-32.f.1.7, 320.192.1-32.f.1.8 |
Cyclic 32-isogeny field degree: |
$1$ |
Cyclic 32-torsion field degree: |
$8$ |
Full 32-torsion field degree: |
$4096$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - x $ |
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.
Maps to other modular curves
$j$-invariant map
of degree 96 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{12x^{2}y^{28}z^{2}+1230x^{2}y^{24}z^{6}-14391x^{2}y^{20}z^{10}+47925x^{2}y^{16}z^{14}+69576x^{2}y^{12}z^{18}-503685x^{2}y^{8}z^{22}-184341x^{2}y^{4}z^{26}-4095x^{2}z^{30}+8xy^{30}z-306xy^{26}z^{5}+1368xy^{22}z^{9}-26537xy^{18}z^{13}+212976xy^{14}z^{17}-430185xy^{10}z^{21}-491500xy^{6}z^{25}-45057xy^{2}z^{29}-y^{32}-168y^{28}z^{4}+3172y^{24}z^{8}-24974y^{20}z^{12}+119316y^{16}z^{16}-155504y^{12}z^{20}-344206y^{8}z^{24}-40938y^{4}z^{28}-z^{32}}{z^{10}y^{8}(x^{2}y^{12}-79x^{2}y^{8}z^{4}-240x^{2}y^{4}z^{8}-16x^{2}z^{12}-13xy^{10}z^{3}-320xy^{6}z^{7}-112xy^{2}z^{11}+2y^{12}z^{2}-160y^{8}z^{6}-96y^{4}z^{10})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.