Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 48 x^{2} + 6 y^{2} + 4 z^{2} + w^{2} $ |
| $=$ | $12 x^{3} + 6 x y^{2} - y z w$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 4 x^{4} y^{2} + 4 x^{2} y^{4} + 12 x^{2} y^{2} z^{2} + 6 y^{4} z^{2} + 9 y^{2} z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^6\cdot3^3\,\frac{3373824xyz^{9}w-10372608xyz^{7}w^{3}+6846912xyz^{5}w^{5}-1335168xyz^{3}w^{7}+82704xyzw^{9}-125184y^{2}z^{10}+1629792y^{2}z^{8}w^{2}-2663280y^{2}z^{6}w^{4}+1202040y^{2}z^{4}w^{6}-186552y^{2}z^{2}w^{8}+10374y^{2}w^{10}-85760z^{12}+548544z^{10}w^{2}-250256z^{8}w^{4}-103640z^{6}w^{6}+21844z^{4}w^{8}+3504z^{2}w^{10}-575w^{12}}{6144xyz^{9}w+4608xyz^{7}w^{3}+1152xyz^{5}w^{5}-2304xyz^{3}w^{7}+240xyzw^{9}+768y^{2}z^{10}+576y^{2}z^{8}w^{2}-1296y^{2}z^{4}w^{6}+252y^{2}z^{2}w^{8}-6y^{2}w^{10}+512z^{12}-768z^{10}w^{2}-736z^{8}w^{4}+32z^{6}w^{6}+80z^{4}w^{8}+6z^{2}w^{10}-w^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
24.72.4.im.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}+4X^{4}Y^{2}+4X^{2}Y^{4}+12X^{2}Y^{2}Z^{2}+6Y^{4}Z^{2}+9Y^{2}Z^{4} $ |
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.
