$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}13&0\\16&19\end{bmatrix}$, $\begin{bmatrix}23&3\\8&11\end{bmatrix}$, $\begin{bmatrix}23&6\\0&7\end{bmatrix}$, $\begin{bmatrix}23&12\\8&17\end{bmatrix}$, $\begin{bmatrix}23&15\\12&19\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1035865 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.3-24.gm.2.1, 24.192.3-24.gm.2.2, 24.192.3-24.gm.2.3, 24.192.3-24.gm.2.4, 24.192.3-24.gm.2.5, 24.192.3-24.gm.2.6, 24.192.3-24.gm.2.7, 24.192.3-24.gm.2.8, 24.192.3-24.gm.2.9, 24.192.3-24.gm.2.10, 24.192.3-24.gm.2.11, 24.192.3-24.gm.2.12, 24.192.3-24.gm.2.13, 24.192.3-24.gm.2.14, 24.192.3-24.gm.2.15, 24.192.3-24.gm.2.16, 120.192.3-24.gm.2.1, 120.192.3-24.gm.2.2, 120.192.3-24.gm.2.3, 120.192.3-24.gm.2.4, 120.192.3-24.gm.2.5, 120.192.3-24.gm.2.6, 120.192.3-24.gm.2.7, 120.192.3-24.gm.2.8, 120.192.3-24.gm.2.9, 120.192.3-24.gm.2.10, 120.192.3-24.gm.2.11, 120.192.3-24.gm.2.12, 120.192.3-24.gm.2.13, 120.192.3-24.gm.2.14, 120.192.3-24.gm.2.15, 120.192.3-24.gm.2.16, 168.192.3-24.gm.2.1, 168.192.3-24.gm.2.2, 168.192.3-24.gm.2.3, 168.192.3-24.gm.2.4, 168.192.3-24.gm.2.5, 168.192.3-24.gm.2.6, 168.192.3-24.gm.2.7, 168.192.3-24.gm.2.8, 168.192.3-24.gm.2.9, 168.192.3-24.gm.2.10, 168.192.3-24.gm.2.11, 168.192.3-24.gm.2.12, 168.192.3-24.gm.2.13, 168.192.3-24.gm.2.14, 168.192.3-24.gm.2.15, 168.192.3-24.gm.2.16, 264.192.3-24.gm.2.1, 264.192.3-24.gm.2.2, 264.192.3-24.gm.2.3, 264.192.3-24.gm.2.4, 264.192.3-24.gm.2.5, 264.192.3-24.gm.2.6, 264.192.3-24.gm.2.7, 264.192.3-24.gm.2.8, 264.192.3-24.gm.2.9, 264.192.3-24.gm.2.10, 264.192.3-24.gm.2.11, 264.192.3-24.gm.2.12, 264.192.3-24.gm.2.13, 264.192.3-24.gm.2.14, 264.192.3-24.gm.2.15, 264.192.3-24.gm.2.16, 312.192.3-24.gm.2.1, 312.192.3-24.gm.2.2, 312.192.3-24.gm.2.3, 312.192.3-24.gm.2.4, 312.192.3-24.gm.2.5, 312.192.3-24.gm.2.6, 312.192.3-24.gm.2.7, 312.192.3-24.gm.2.8, 312.192.3-24.gm.2.9, 312.192.3-24.gm.2.10, 312.192.3-24.gm.2.11, 312.192.3-24.gm.2.12, 312.192.3-24.gm.2.13, 312.192.3-24.gm.2.14, 312.192.3-24.gm.2.15, 312.192.3-24.gm.2.16 |
Cyclic 24-isogeny field degree: |
$2$ |
Cyclic 24-torsion field degree: |
$16$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x y t - x w t - y^{2} t + 2 y z t - 2 z^{2} t + z w t $ |
| $=$ | $2 x y t + x w t - 2 y z t - y w t$ |
| $=$ | $x y t + 2 x w t - y^{2} t - y w t + 2 z^{2} t + z w t - w^{2} t$ |
| $=$ | $2 x^{3} - 2 x y^{2} + x t^{2} - 2 y t^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{7} + 6 x^{6} z + 2 x^{5} y^{2} + 16 x^{5} z^{2} + 12 x^{4} y^{2} z + 24 x^{4} z^{3} + \cdots + 4 y^{2} z^{5} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -2x^{7} - 10x^{6} - 14x^{5} - 20x^{4} - 14x^{3} - 10x^{2} - 2x $ |
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 between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4y^{3}t+8y^{2}wt+5yw^{2}t+w^{3}t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle -y-w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{43322x^{2}w^{12}-56884x^{2}w^{10}t^{2}+4316x^{2}w^{8}t^{4}+44520x^{2}w^{6}t^{6}-5586x^{2}w^{4}t^{8}+684x^{2}w^{2}t^{10}-158x^{2}t^{12}-41850xzw^{12}+54516xzw^{10}t^{2}+77920xzw^{8}t^{4}-62056xzw^{6}t^{6}+7026xzw^{4}t^{8}-2628xzw^{2}t^{10}+1134xzt^{12}-19901xw^{13}-79625xw^{11}t^{2}-31836xw^{9}t^{4}-21776xw^{7}t^{6}+21081xw^{5}t^{8}-6369xw^{3}t^{10}+1697xwt^{12}+20736yw^{13}+10686yw^{11}t^{2}-13860yw^{9}t^{4}-20404yw^{7}t^{6}+2677yw^{5}t^{8}-5514yw^{3}t^{10}+510ywt^{12}+2428z^{2}w^{12}-122448z^{2}w^{10}t^{2}-108224z^{2}w^{8}t^{4}+10240z^{2}w^{6}t^{6}+9764z^{2}w^{4}t^{8}+1248z^{2}w^{2}t^{10}-972z^{2}t^{12}-256zw^{13}-37700zw^{11}t^{2}-14876zw^{9}t^{4}+33152zw^{7}t^{6}-16984zw^{5}t^{8}+16860zw^{3}t^{10}-2920zwt^{12}-735w^{14}+85998w^{12}t^{2}-31078w^{10}t^{4}-3202w^{8}t^{6}-12656w^{6}t^{8}+9870w^{4}t^{10}-1019w^{2}t^{12}+2t^{14}}{t^{2}w^{3}(4x^{2}w^{7}+2x^{2}w^{5}t^{2}-120x^{2}w^{3}t^{4}-122x^{2}wt^{6}-4xzw^{7}+42xzw^{5}t^{2}+120xzw^{3}t^{4}+122xzwt^{6}-41xw^{6}t^{2}-48xw^{4}t^{4}+477xw^{2}t^{6}-81xt^{8}-2yw^{6}t^{2}+18yw^{4}t^{4}+57yw^{2}t^{6}-122yt^{8}-84z^{2}w^{5}t^{2}+244z^{2}wt^{6}+4zw^{6}t^{2}-344zw^{2}t^{6}+324zt^{8}+21w^{7}t^{2}-20w^{5}t^{4}-176w^{3}t^{6}+202wt^{8})}$ |
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.