Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 2 x^{2} + y z + w^{2} $ |
| $=$ | $2 x^{2} w + 4 x y^{2} + x z^{2} - 3 y z w - 2 w^{3}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{6} + 3 x^{4} y z + 2 x^{2} y^{4} - x^{2} y^{2} z^{2} + 2 x^{2} z^{4} + y^{3} z^{3} $ |
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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 to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{4608xyz^{7}w^{3}-39848xyz^{3}w^{7}+4093xz^{10}w-51728xz^{6}w^{5}+95592xz^{2}w^{9}+512y^{12}-768y^{8}w^{4}+288y^{4}w^{8}+2048y^{2}z^{10}-26052y^{2}z^{6}w^{4}+31616y^{2}z^{2}w^{8}+45yz^{9}w^{2}-11008yz^{5}w^{6}-2376yzw^{10}+512z^{12}-10198z^{8}w^{4}+40706z^{4}w^{8}-33912w^{12}}{w^{3}(xyz^{7}-56xyz^{3}w^{4}+7xz^{6}w^{2}-30xz^{2}w^{6}-6y^{2}z^{6}w+8y^{2}z^{2}w^{5}-37yz^{5}w^{3}+12yzw^{7}-25z^{4}w^{5}+2w^{9})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
48.72.4.w.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{6}+3X^{4}YZ+2X^{2}Y^{4}-X^{2}Y^{2}Z^{2}+2X^{2}Z^{4}+Y^{3}Z^{3} $ |
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.
