$\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}1&1\\0&7\end{bmatrix}$, $\begin{bmatrix}7&1\\6&11\end{bmatrix}$, $\begin{bmatrix}7&3\\6&11\end{bmatrix}$, $\begin{bmatrix}11&2\\0&11\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: |
$C_2^3:D_6$ |
Contains $-I$: |
yes |
Quadratic refinements: |
12.96.1-12.g.1.1, 12.96.1-12.g.1.2, 12.96.1-12.g.1.3, 12.96.1-12.g.1.4, 12.96.1-12.g.1.5, 12.96.1-12.g.1.6, 24.96.1-12.g.1.1, 24.96.1-12.g.1.2, 24.96.1-12.g.1.3, 24.96.1-12.g.1.4, 24.96.1-12.g.1.5, 24.96.1-12.g.1.6, 60.96.1-12.g.1.1, 60.96.1-12.g.1.2, 60.96.1-12.g.1.3, 60.96.1-12.g.1.4, 60.96.1-12.g.1.5, 60.96.1-12.g.1.6, 84.96.1-12.g.1.1, 84.96.1-12.g.1.2, 84.96.1-12.g.1.3, 84.96.1-12.g.1.4, 84.96.1-12.g.1.5, 84.96.1-12.g.1.6, 120.96.1-12.g.1.1, 120.96.1-12.g.1.2, 120.96.1-12.g.1.3, 120.96.1-12.g.1.4, 120.96.1-12.g.1.5, 120.96.1-12.g.1.6, 132.96.1-12.g.1.1, 132.96.1-12.g.1.2, 132.96.1-12.g.1.3, 132.96.1-12.g.1.4, 132.96.1-12.g.1.5, 132.96.1-12.g.1.6, 156.96.1-12.g.1.1, 156.96.1-12.g.1.2, 156.96.1-12.g.1.3, 156.96.1-12.g.1.4, 156.96.1-12.g.1.5, 156.96.1-12.g.1.6, 168.96.1-12.g.1.1, 168.96.1-12.g.1.2, 168.96.1-12.g.1.3, 168.96.1-12.g.1.4, 168.96.1-12.g.1.5, 168.96.1-12.g.1.6, 204.96.1-12.g.1.1, 204.96.1-12.g.1.2, 204.96.1-12.g.1.3, 204.96.1-12.g.1.4, 204.96.1-12.g.1.5, 204.96.1-12.g.1.6, 228.96.1-12.g.1.1, 228.96.1-12.g.1.2, 228.96.1-12.g.1.3, 228.96.1-12.g.1.4, 228.96.1-12.g.1.5, 228.96.1-12.g.1.6, 264.96.1-12.g.1.1, 264.96.1-12.g.1.2, 264.96.1-12.g.1.3, 264.96.1-12.g.1.4, 264.96.1-12.g.1.5, 264.96.1-12.g.1.6, 276.96.1-12.g.1.1, 276.96.1-12.g.1.2, 276.96.1-12.g.1.3, 276.96.1-12.g.1.4, 276.96.1-12.g.1.5, 276.96.1-12.g.1.6, 312.96.1-12.g.1.1, 312.96.1-12.g.1.2, 312.96.1-12.g.1.3, 312.96.1-12.g.1.4, 312.96.1-12.g.1.5, 312.96.1-12.g.1.6 |
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} $ |
| $=$ | $x^{2} - x y + 9 y^{2} + 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} + 3 x^{2} y^{2} + 10 x^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3}\cdot\frac{(4z^{2}+3w^{2})(279552y^{2}z^{8}-39168y^{2}z^{6}w^{2}-98496y^{2}z^{4}w^{4}-589680y^{2}z^{2}w^{6}-176904y^{2}w^{8}+10240z^{10}-9216z^{8}w^{2}-46656z^{6}w^{4}-212112z^{4}w^{6}-131220z^{2}w^{8}-19683w^{10})}{w^{2}z^{4}(24y^{2}z^{4}-18y^{2}z^{2}w^{2}-27y^{2}w^{4}+8z^{6}-3z^{4}w^{2})}$ |
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.