| $\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}5&1\\8&5\end{bmatrix}$, $\begin{bmatrix}5&10\\4&1\end{bmatrix}$, $\begin{bmatrix}7&7\\8&7\end{bmatrix}$, $\begin{bmatrix}11&10\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: |
$C_2^3:\SD_{16}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
12.72.2-12.t.1.1, 12.72.2-12.t.1.2, 12.72.2-12.t.1.3, 12.72.2-12.t.1.4, 24.72.2-12.t.1.1, 24.72.2-12.t.1.2, 24.72.2-12.t.1.3, 24.72.2-12.t.1.4, 24.72.2-12.t.1.5, 24.72.2-12.t.1.6, 24.72.2-12.t.1.7, 24.72.2-12.t.1.8, 24.72.2-12.t.1.9, 24.72.2-12.t.1.10, 24.72.2-12.t.1.11, 24.72.2-12.t.1.12, 60.72.2-12.t.1.1, 60.72.2-12.t.1.2, 60.72.2-12.t.1.3, 60.72.2-12.t.1.4, 84.72.2-12.t.1.1, 84.72.2-12.t.1.2, 84.72.2-12.t.1.3, 84.72.2-12.t.1.4, 120.72.2-12.t.1.1, 120.72.2-12.t.1.2, 120.72.2-12.t.1.3, 120.72.2-12.t.1.4, 120.72.2-12.t.1.5, 120.72.2-12.t.1.6, 120.72.2-12.t.1.7, 120.72.2-12.t.1.8, 120.72.2-12.t.1.9, 120.72.2-12.t.1.10, 120.72.2-12.t.1.11, 120.72.2-12.t.1.12, 132.72.2-12.t.1.1, 132.72.2-12.t.1.2, 132.72.2-12.t.1.3, 132.72.2-12.t.1.4, 156.72.2-12.t.1.1, 156.72.2-12.t.1.2, 156.72.2-12.t.1.3, 156.72.2-12.t.1.4, 168.72.2-12.t.1.1, 168.72.2-12.t.1.2, 168.72.2-12.t.1.3, 168.72.2-12.t.1.4, 168.72.2-12.t.1.5, 168.72.2-12.t.1.6, 168.72.2-12.t.1.7, 168.72.2-12.t.1.8, 168.72.2-12.t.1.9, 168.72.2-12.t.1.10, 168.72.2-12.t.1.11, 168.72.2-12.t.1.12, 204.72.2-12.t.1.1, 204.72.2-12.t.1.2, 204.72.2-12.t.1.3, 204.72.2-12.t.1.4, 228.72.2-12.t.1.1, 228.72.2-12.t.1.2, 228.72.2-12.t.1.3, 228.72.2-12.t.1.4, 264.72.2-12.t.1.1, 264.72.2-12.t.1.2, 264.72.2-12.t.1.3, 264.72.2-12.t.1.4, 264.72.2-12.t.1.5, 264.72.2-12.t.1.6, 264.72.2-12.t.1.7, 264.72.2-12.t.1.8, 264.72.2-12.t.1.9, 264.72.2-12.t.1.10, 264.72.2-12.t.1.11, 264.72.2-12.t.1.12, 276.72.2-12.t.1.1, 276.72.2-12.t.1.2, 276.72.2-12.t.1.3, 276.72.2-12.t.1.4, 312.72.2-12.t.1.1, 312.72.2-12.t.1.2, 312.72.2-12.t.1.3, 312.72.2-12.t.1.4, 312.72.2-12.t.1.5, 312.72.2-12.t.1.6, 312.72.2-12.t.1.7, 312.72.2-12.t.1.8, 312.72.2-12.t.1.9, 312.72.2-12.t.1.10, 312.72.2-12.t.1.11, 312.72.2-12.t.1.12 |
| Cyclic 12-isogeny field degree: |
$4$ |
| Cyclic 12-torsion field degree: |
$16$ |
| Full 12-torsion field degree: |
$128$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} w + y z w $ |
| $=$ | $3 x^{2} z + y z^{2}$ |
| $=$ | $3 x^{2} y + y^{2} z$ |
| $=$ | $3 x^{3} + x y z$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - 27 x y^{2} z^{2} + 2 y z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} + 27 $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{18}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{8}xzw-z^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{2}x$ |
Maps to other modular curves
$j$-invariant map
of degree 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^4}\cdot\frac{36912xyz^{4}w^{2}-4096xz^{7}-20736xzw^{6}+14589y^{2}z^{3}w^{3}-12032yz^{6}w-432yw^{7}+73737z^{3}w^{5}}{wz^{6}y}$ |
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.
