$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&14\\16&11\end{bmatrix}$, $\begin{bmatrix}5&16\\0&7\end{bmatrix}$, $\begin{bmatrix}21&14\\20&1\end{bmatrix}$, $\begin{bmatrix}23&0\\16&23\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
$C_2\times D_4\times \GL(2,3)$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.bj.1.1, 24.192.1-24.bj.1.2, 24.192.1-24.bj.1.3, 24.192.1-24.bj.1.4, 24.192.1-24.bj.1.5, 24.192.1-24.bj.1.6, 24.192.1-24.bj.1.7, 24.192.1-24.bj.1.8, 120.192.1-24.bj.1.1, 120.192.1-24.bj.1.2, 120.192.1-24.bj.1.3, 120.192.1-24.bj.1.4, 120.192.1-24.bj.1.5, 120.192.1-24.bj.1.6, 120.192.1-24.bj.1.7, 120.192.1-24.bj.1.8, 168.192.1-24.bj.1.1, 168.192.1-24.bj.1.2, 168.192.1-24.bj.1.3, 168.192.1-24.bj.1.4, 168.192.1-24.bj.1.5, 168.192.1-24.bj.1.6, 168.192.1-24.bj.1.7, 168.192.1-24.bj.1.8, 264.192.1-24.bj.1.1, 264.192.1-24.bj.1.2, 264.192.1-24.bj.1.3, 264.192.1-24.bj.1.4, 264.192.1-24.bj.1.5, 264.192.1-24.bj.1.6, 264.192.1-24.bj.1.7, 264.192.1-24.bj.1.8, 312.192.1-24.bj.1.1, 312.192.1-24.bj.1.2, 312.192.1-24.bj.1.3, 312.192.1-24.bj.1.4, 312.192.1-24.bj.1.5, 312.192.1-24.bj.1.6, 312.192.1-24.bj.1.7, 312.192.1-24.bj.1.8 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 y^{2} + 2 w^{2} $ |
| $=$ | $x^{2} - 2 x z + x w - 2 y^{2} + 2 z^{2} - 2 z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 4 x^{3} z - 4 x^{2} y^{2} + 2 x^{2} z^{2} + 4 x y^{2} z + 8 y^{4} + 9 y^{2} z^{2} + 3 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{50331648xz^{21}w^{2}-528482304xz^{20}w^{3}+2774532096xz^{19}w^{4}-9622781952xz^{18}w^{5}+24475336704xz^{17}w^{6}-48009314304xz^{16}w^{7}+74406494208xz^{15}w^{8}-91672510464xz^{14}w^{9}+88478367744xz^{13}w^{10}-63135621120xz^{12}w^{11}+26228920320xz^{11}w^{12}+6403491840xz^{10}w^{13}-24137401344xz^{9}w^{14}+26457933312xz^{8}w^{15}-19797432960xz^{7}w^{16}+11303098560xz^{6}w^{17}-5010238944xz^{5}w^{18}+1675954896xz^{4}w^{19}-376483752xz^{3}w^{20}+30863628xz^{2}w^{21}+19026330xzw^{22}-7042071xw^{23}-8388608z^{24}+100663296z^{23}w-629145600z^{22}w^{2}+2675965952z^{21}w^{3}-8477736960z^{20}w^{4}+20831010816z^{19}w^{5}-40363884544z^{18}w^{6}+61502128128z^{17}w^{7}-71282884608z^{16}w^{8}+55454203904z^{15}w^{9}-10094690304z^{14}w^{10}-51028475904z^{13}w^{11}+102416744448z^{12}w^{12}-123259404288z^{11}w^{13}+110407351296z^{10}w^{14}-77352152064z^{9}w^{15}+42296006784z^{8}w^{16}-17007942144z^{7}w^{17}+3806245344z^{6}w^{18}+796031328z^{5}w^{19}-1340638608z^{4}w^{20}+783063680z^{3}w^{21}-303405030z^{2}w^{22}+79333686zw^{23}-17267750w^{24}}{w^{8}(32768xz^{15}-245760xz^{14}w+843776xz^{13}w^{2}-1757184xz^{12}w^{3}+2242560xz^{11}w^{4}-1205248xz^{10}w^{5}-1627648xz^{9}w^{6}+4942080xz^{8}w^{7}-6679936xz^{7}w^{8}+5937472xz^{6}w^{9}-3652512xz^{5}w^{10}+1488624xz^{4}w^{11}-326472xz^{3}w^{12}-7524xz^{2}w^{13}+20938xzw^{14}-2967xw^{15}+139264z^{14}w^{2}-974848z^{13}w^{3}+3440640z^{12}w^{4}-7970816z^{11}w^{5}+13240832z^{10}w^{6}-16372224z^{9}w^{7}+15110272z^{8}w^{8}-10057216z^{7}w^{9}+4240800z^{6}w^{10}-427488z^{5}w^{11}-810096z^{4}w^{12}+625920z^{3}w^{13}-221926z^{2}w^{14}+36886zw^{15}-1830w^{16})}$ |
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.