$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&4\\16&5\end{bmatrix}$, $\begin{bmatrix}9&20\\16&5\end{bmatrix}$, $\begin{bmatrix}11&10\\0&7\end{bmatrix}$, $\begin{bmatrix}13&6\\8&17\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1089047 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.cc.1.1, 24.192.1-24.cc.1.2, 24.192.1-24.cc.1.3, 24.192.1-24.cc.1.4, 24.192.1-24.cc.1.5, 24.192.1-24.cc.1.6, 24.192.1-24.cc.1.7, 24.192.1-24.cc.1.8, 48.192.1-24.cc.1.1, 48.192.1-24.cc.1.2, 48.192.1-24.cc.1.3, 48.192.1-24.cc.1.4, 120.192.1-24.cc.1.1, 120.192.1-24.cc.1.2, 120.192.1-24.cc.1.3, 120.192.1-24.cc.1.4, 120.192.1-24.cc.1.5, 120.192.1-24.cc.1.6, 120.192.1-24.cc.1.7, 120.192.1-24.cc.1.8, 168.192.1-24.cc.1.1, 168.192.1-24.cc.1.2, 168.192.1-24.cc.1.3, 168.192.1-24.cc.1.4, 168.192.1-24.cc.1.5, 168.192.1-24.cc.1.6, 168.192.1-24.cc.1.7, 168.192.1-24.cc.1.8, 240.192.1-24.cc.1.1, 240.192.1-24.cc.1.2, 240.192.1-24.cc.1.3, 240.192.1-24.cc.1.4, 264.192.1-24.cc.1.1, 264.192.1-24.cc.1.2, 264.192.1-24.cc.1.3, 264.192.1-24.cc.1.4, 264.192.1-24.cc.1.5, 264.192.1-24.cc.1.6, 264.192.1-24.cc.1.7, 264.192.1-24.cc.1.8, 312.192.1-24.cc.1.1, 312.192.1-24.cc.1.2, 312.192.1-24.cc.1.3, 312.192.1-24.cc.1.4, 312.192.1-24.cc.1.5, 312.192.1-24.cc.1.6, 312.192.1-24.cc.1.7, 312.192.1-24.cc.1.8 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 y^{2} - y z - y w + z^{2} + w^{2} $ |
| $=$ | $x^{2} + y^{2} + y z + y w + z^{2} - 4 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} - 16 x^{3} z + 12 x^{2} z^{2} - 4 x z^{3} + y^{4} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^2}\cdot\frac{219585645yz^{23}+772555779yz^{22}w-9507399183yz^{21}w^{2}+23197051359yz^{20}w^{3}-157603401189yz^{19}w^{4}+294109531893yz^{18}w^{5}-867954960081yz^{17}w^{6}+1367086692609yz^{16}w^{7}-2107729320894yz^{15}w^{8}+2361853727262yz^{14}w^{9}-1776427047990yz^{13}w^{10}+871982984790yz^{12}w^{11}+871982984790yz^{11}w^{12}-1776427047990yz^{10}w^{13}+2361853727262yz^{9}w^{14}-2107729320894yz^{8}w^{15}+1367086692609yz^{7}w^{16}-867954960081yz^{6}w^{17}+294109531893yz^{5}w^{18}-157603401189yz^{4}w^{19}+23197051359yz^{3}w^{20}-9507399183yz^{2}w^{21}+772555779yzw^{22}+219585645yw^{23}+100396841z^{24}-2787859614z^{23}w+10112374488z^{22}w^{2}-47495241202z^{21}w^{3}+186567415914z^{20}w^{4}-464328212250z^{19}w^{5}+1432113620792z^{18}w^{6}-2731880479158z^{17}w^{7}+5878669139559z^{16}w^{8}-8899838705548z^{15}w^{9}+13783904928240z^{14}w^{10}-16094086106388z^{13}w^{11}+18355947680780z^{12}w^{12}-16094086106388z^{11}w^{13}+13783904928240z^{10}w^{14}-8899838705548z^{9}w^{15}+5878669139559z^{8}w^{16}-2731880479158z^{7}w^{17}+1432113620792z^{6}w^{18}-464328212250z^{5}w^{19}+186567415914z^{4}w^{20}-47495241202z^{3}w^{21}+10112374488z^{2}w^{22}-2787859614zw^{23}+100396841w^{24}}{(z-w)^{8}(14805yz^{15}+147155yz^{14}w-32203yz^{13}w^{2}-446589yz^{12}w^{3}-942935yz^{11}w^{4}-2860481yz^{10}w^{5}-3228039yz^{9}w^{6}-4595649yz^{8}w^{7}-4595649yz^{7}w^{8}-3228039yz^{6}w^{9}-2860481yz^{5}w^{10}-942935yz^{4}w^{11}-446589yz^{3}w^{12}-32203yz^{2}w^{13}+147155yzw^{14}+14805yw^{15}+6769z^{16}-96566z^{15}w-353620z^{14}w^{2}-264210z^{13}w^{3}-700956z^{12}w^{4}-121478z^{11}w^{5}+456020z^{10}w^{6}+277454z^{9}w^{7}+1593174z^{8}w^{8}+277454z^{7}w^{9}+456020z^{6}w^{10}-121478z^{5}w^{11}-700956z^{4}w^{12}-264210z^{3}w^{13}-353620z^{2}w^{14}-96566zw^{15}+6769w^{16})}$ |
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.