$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}5&4\\0&9\end{bmatrix}$, $\begin{bmatrix}11&4\\0&15\end{bmatrix}$, $\begin{bmatrix}11&5\\0&13\end{bmatrix}$, $\begin{bmatrix}15&6\\0&15\end{bmatrix}$ |
$\GL_2(\Z/16\Z)$-subgroup: |
$C_2\times C_8.C_4^2$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.192.1-16.o.2.1, 16.192.1-16.o.2.2, 16.192.1-16.o.2.3, 16.192.1-16.o.2.4, 16.192.1-16.o.2.5, 16.192.1-16.o.2.6, 32.192.1-16.o.2.1, 32.192.1-16.o.2.2, 32.192.1-16.o.2.3, 32.192.1-16.o.2.4, 48.192.1-16.o.2.1, 48.192.1-16.o.2.2, 48.192.1-16.o.2.3, 48.192.1-16.o.2.4, 48.192.1-16.o.2.5, 48.192.1-16.o.2.6, 80.192.1-16.o.2.1, 80.192.1-16.o.2.2, 80.192.1-16.o.2.3, 80.192.1-16.o.2.4, 80.192.1-16.o.2.5, 80.192.1-16.o.2.6, 96.192.1-16.o.2.1, 96.192.1-16.o.2.2, 96.192.1-16.o.2.3, 96.192.1-16.o.2.4, 112.192.1-16.o.2.1, 112.192.1-16.o.2.2, 112.192.1-16.o.2.3, 112.192.1-16.o.2.4, 112.192.1-16.o.2.5, 112.192.1-16.o.2.6, 160.192.1-16.o.2.1, 160.192.1-16.o.2.2, 160.192.1-16.o.2.3, 160.192.1-16.o.2.4, 176.192.1-16.o.2.1, 176.192.1-16.o.2.2, 176.192.1-16.o.2.3, 176.192.1-16.o.2.4, 176.192.1-16.o.2.5, 176.192.1-16.o.2.6, 208.192.1-16.o.2.1, 208.192.1-16.o.2.2, 208.192.1-16.o.2.3, 208.192.1-16.o.2.4, 208.192.1-16.o.2.5, 208.192.1-16.o.2.6, 224.192.1-16.o.2.1, 224.192.1-16.o.2.2, 224.192.1-16.o.2.3, 224.192.1-16.o.2.4, 240.192.1-16.o.2.1, 240.192.1-16.o.2.2, 240.192.1-16.o.2.3, 240.192.1-16.o.2.4, 240.192.1-16.o.2.5, 240.192.1-16.o.2.6, 272.192.1-16.o.2.1, 272.192.1-16.o.2.2, 272.192.1-16.o.2.3, 272.192.1-16.o.2.4, 272.192.1-16.o.2.5, 272.192.1-16.o.2.6, 304.192.1-16.o.2.1, 304.192.1-16.o.2.2, 304.192.1-16.o.2.3, 304.192.1-16.o.2.4, 304.192.1-16.o.2.5, 304.192.1-16.o.2.6 |
Cyclic 16-isogeny field degree: |
$1$ |
Cyclic 16-torsion field degree: |
$8$ |
Full 16-torsion field degree: |
$256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x z + y^{2} - y w $ |
| $=$ | $x^{2} - y^{2} + 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{3} y + x^{2} y^{2} - x^{2} z^{2} + 2 y^{2} z^{2} + 2 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{13696215750xyz^{21}w+53416636812xyz^{19}w^{3}+67637157318xyz^{17}w^{5}+11901298256xyz^{15}w^{7}-42126311220xyz^{13}w^{9}-34542875256xyz^{11}w^{11}-8291321348xyz^{9}w^{13}+542183888xyz^{7}w^{15}+428347566xyz^{5}w^{17}+24114380xyz^{3}w^{19}-2621442xyzw^{21}-1409416875xz^{23}-31075931745xz^{21}w^{2}-109083358233xz^{19}w^{4}-141823236203xz^{17}w^{6}-49515974974xz^{15}w^{8}+49457701830xz^{13}w^{10}+49776957758xz^{11}w^{12}+13549082170xz^{9}w^{14}-149350407xz^{7}w^{16}-530152245xz^{5}w^{18}-38534453xz^{3}w^{20}+2883585xzw^{22}+2035217475yz^{22}w-23985304335yz^{20}w^{3}-130735876815yz^{18}w^{5}-236605158101yz^{16}w^{7}-190853329554yz^{14}w^{9}-56100788838yz^{12}w^{11}+10946882738yz^{10}w^{13}+9545627270yz^{8}w^{15}+1410166591yz^{6}w^{17}-72787515yz^{4}w^{19}-21234387yz^{2}w^{21}+262143yw^{23}-8987105250z^{24}-33157678050z^{22}w^{2}+10539624114z^{20}w^{4}+212529468450z^{18}w^{6}+392929104172z^{16}w^{8}+312077226732z^{14}w^{10}+99340674852z^{12}w^{12}-6877841596z^{10}w^{14}-11076743978z^{8}w^{16}-1785516650z^{6}w^{18}+63955978z^{4}w^{20}+23592954z^{2}w^{22}-262144w^{24}}{z^{2}(z^{2}+w^{2})(1852xyz^{17}w-29344xyz^{15}w^{3}-543824xyz^{13}w^{5}-2090624xyz^{11}w^{7}-3243416xyz^{9}w^{9}-1928608xyz^{7}w^{11}+125104xyz^{5}w^{13}+393408xyz^{3}w^{15}-4xyzw^{17}-85xz^{19}+18921xz^{17}w^{2}+460380xz^{15}w^{4}+2690324xz^{13}w^{6}+6885978xz^{11}w^{8}+8912222xz^{9}w^{10}+5385420xz^{7}w^{12}+657060xz^{5}w^{14}-458797xz^{3}w^{16}+xzw^{18}+6665yz^{18}w+169719yz^{16}w^{3}+965268yz^{14}w^{5}+2710156yz^{12}w^{7}+5286046yz^{10}w^{9}+7218626yz^{8}w^{11}+5603236yz^{6}w^{13}+1702844yz^{4}w^{15}-65503yz^{2}w^{17}-yw^{19}-542z^{20}-35722z^{18}w^{2}-241512z^{16}w^{4}-596344z^{14}w^{6}-1489780z^{12}w^{8}-4545276z^{10}w^{10}-8115944z^{8}w^{12}-6766072z^{6}w^{14}-2031134z^{4}w^{16}+65526z^{2}w^{18})}$ |
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.