$\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}5&4\\0&11\end{bmatrix}$, $\begin{bmatrix}11&1\\6&1\end{bmatrix}$, $\begin{bmatrix}11&8\\0&11\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: |
$C_2^3.D_6$ |
Contains $-I$: |
yes |
Quadratic refinements: |
12.96.1-12.m.1.1, 12.96.1-12.m.1.2, 12.96.1-12.m.1.3, 12.96.1-12.m.1.4, 24.96.1-12.m.1.1, 24.96.1-12.m.1.2, 24.96.1-12.m.1.3, 24.96.1-12.m.1.4, 24.96.1-12.m.1.5, 24.96.1-12.m.1.6, 24.96.1-12.m.1.7, 24.96.1-12.m.1.8, 60.96.1-12.m.1.1, 60.96.1-12.m.1.2, 60.96.1-12.m.1.3, 60.96.1-12.m.1.4, 84.96.1-12.m.1.1, 84.96.1-12.m.1.2, 84.96.1-12.m.1.3, 84.96.1-12.m.1.4, 120.96.1-12.m.1.1, 120.96.1-12.m.1.2, 120.96.1-12.m.1.3, 120.96.1-12.m.1.4, 120.96.1-12.m.1.5, 120.96.1-12.m.1.6, 120.96.1-12.m.1.7, 120.96.1-12.m.1.8, 132.96.1-12.m.1.1, 132.96.1-12.m.1.2, 132.96.1-12.m.1.3, 132.96.1-12.m.1.4, 156.96.1-12.m.1.1, 156.96.1-12.m.1.2, 156.96.1-12.m.1.3, 156.96.1-12.m.1.4, 168.96.1-12.m.1.1, 168.96.1-12.m.1.2, 168.96.1-12.m.1.3, 168.96.1-12.m.1.4, 168.96.1-12.m.1.5, 168.96.1-12.m.1.6, 168.96.1-12.m.1.7, 168.96.1-12.m.1.8, 204.96.1-12.m.1.1, 204.96.1-12.m.1.2, 204.96.1-12.m.1.3, 204.96.1-12.m.1.4, 228.96.1-12.m.1.1, 228.96.1-12.m.1.2, 228.96.1-12.m.1.3, 228.96.1-12.m.1.4, 264.96.1-12.m.1.1, 264.96.1-12.m.1.2, 264.96.1-12.m.1.3, 264.96.1-12.m.1.4, 264.96.1-12.m.1.5, 264.96.1-12.m.1.6, 264.96.1-12.m.1.7, 264.96.1-12.m.1.8, 276.96.1-12.m.1.1, 276.96.1-12.m.1.2, 276.96.1-12.m.1.3, 276.96.1-12.m.1.4, 312.96.1-12.m.1.1, 312.96.1-12.m.1.2, 312.96.1-12.m.1.3, 312.96.1-12.m.1.4, 312.96.1-12.m.1.5, 312.96.1-12.m.1.6, 312.96.1-12.m.1.7, 312.96.1-12.m.1.8 |
Cyclic 12-isogeny field degree: |
$2$ |
Cyclic 12-torsion field degree: |
$8$ |
Full 12-torsion field degree: |
$96$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x y + z^{2} $ |
| $=$ | $x^{2} - 2 x y - x w + 7 y^{2} + y w + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 63 x^{4} + 3 x^{3} y + x^{2} y^{2} + 24 x^{2} z^{2} + x y z^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^2\cdot7^4}\cdot\frac{380307336192xz^{10}w+1579111236864xz^{8}w^{3}+2521673985600xz^{6}w^{5}+1032544204272xz^{4}w^{7}+69725448000xz^{2}w^{9}+4324931057664y^{2}z^{10}+2359478757888y^{2}z^{8}w^{2}-8652348706176y^{2}z^{6}w^{4}-17162385922080y^{2}z^{4}w^{6}-8638269566976y^{2}z^{2}w^{8}-1445995352880y^{2}w^{10}-380307336192yz^{10}w-1896310612224yz^{8}w^{3}-2107234011456yz^{6}w^{5}-1197167147568yz^{4}w^{7}-772917895440yz^{2}w^{9}-209176344000yw^{11}+284625065984z^{12}-228213167616z^{10}w^{2}-3221208337536z^{8}w^{4}-7507988125344z^{6}w^{6}-5563023611040z^{4}w^{8}-1717121100168z^{2}w^{10}-205924456521w^{12}}{z^{4}(4900xz^{6}w-3535xz^{4}w^{3}+3240xz^{2}w^{5}+8232y^{2}z^{6}-31458y^{2}z^{4}w^{2}+33726y^{2}z^{2}w^{4}+6723y^{2}w^{6}-4900yz^{6}w+6671yz^{4}w^{3}-9834yz^{2}w^{5}-9720yw^{7}+2744z^{8}-9114z^{6}w^{2}+5362z^{4}w^{4}-3240z^{2}w^{6})}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.