$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&10\\8&21\end{bmatrix}$, $\begin{bmatrix}3&2\\16&7\end{bmatrix}$, $\begin{bmatrix}7&12\\0&11\end{bmatrix}$, $\begin{bmatrix}17&11\\0&23\end{bmatrix}$, $\begin{bmatrix}17&17\\16&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.dk.1.1, 24.96.1-24.dk.1.2, 24.96.1-24.dk.1.3, 24.96.1-24.dk.1.4, 24.96.1-24.dk.1.5, 24.96.1-24.dk.1.6, 48.96.1-24.dk.1.1, 48.96.1-24.dk.1.2, 48.96.1-24.dk.1.3, 48.96.1-24.dk.1.4, 48.96.1-24.dk.1.5, 48.96.1-24.dk.1.6, 120.96.1-24.dk.1.1, 120.96.1-24.dk.1.2, 120.96.1-24.dk.1.3, 120.96.1-24.dk.1.4, 120.96.1-24.dk.1.5, 120.96.1-24.dk.1.6, 168.96.1-24.dk.1.1, 168.96.1-24.dk.1.2, 168.96.1-24.dk.1.3, 168.96.1-24.dk.1.4, 168.96.1-24.dk.1.5, 168.96.1-24.dk.1.6, 240.96.1-24.dk.1.1, 240.96.1-24.dk.1.2, 240.96.1-24.dk.1.3, 240.96.1-24.dk.1.4, 240.96.1-24.dk.1.5, 240.96.1-24.dk.1.6, 264.96.1-24.dk.1.1, 264.96.1-24.dk.1.2, 264.96.1-24.dk.1.3, 264.96.1-24.dk.1.4, 264.96.1-24.dk.1.5, 264.96.1-24.dk.1.6, 312.96.1-24.dk.1.1, 312.96.1-24.dk.1.2, 312.96.1-24.dk.1.3, 312.96.1-24.dk.1.4, 312.96.1-24.dk.1.5, 312.96.1-24.dk.1.6 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$1536$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 36x $ |
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map
of degree 48 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3^2}\cdot\frac{12921120x^{2}y^{12}z^{2}+5923931728896x^{2}y^{8}z^{6}-51499361360609280x^{2}y^{4}z^{10}+538990877234083921920x^{2}z^{14}-6192xy^{14}z+271155527424xy^{10}z^{5}-8840731536654336xy^{6}z^{9}+74881781010929811456xy^{2}z^{13}+y^{16}-9374683392y^{12}z^{4}-155483208695808y^{8}z^{8}+1645880657768349696y^{4}z^{12}+4738381338321616896z^{16}}{zy^{4}(468x^{2}y^{8}z+841487616x^{2}y^{4}z^{5}+19982861844480x^{2}z^{9}+xy^{10}+12503808xy^{6}z^{4}+1673945616384xy^{2}z^{8}+90720y^{8}z^{3}+31563343872y^{4}z^{7}+2821109907456z^{11})}$ |
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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.