Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 12 x^{2} + 6 y^{2} - z w $ |
| $=$ | $12 x^{2} y + 4 x z^{2} + x w^{2} - 12 y^{3} + 3 y z w$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} + 2 x^{4} y z - 16 x^{2} y^{4} + 2 x^{2} y^{2} z^{2} - x^{2} z^{4} + 6 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\cdot3^3\,\frac{62592xyz^{10}-814896xyz^{8}w^{2}+1331640xyz^{6}w^{4}-601020xyz^{4}w^{6}+93276xyz^{2}w^{8}-5187xyw^{10}+210864y^{2}z^{9}w-648288y^{2}z^{7}w^{3}+427932y^{2}z^{5}w^{5}-83448y^{2}z^{3}w^{7}+5169y^{2}zw^{9}-144z^{12}-32320z^{10}w^{2}+78352z^{8}w^{4}-28820z^{6}w^{6}-3383z^{4}w^{8}+1730z^{2}w^{10}-144w^{12}}{384xyz^{10}+288xyz^{8}w^{2}-648xyz^{4}w^{6}+126xyz^{2}w^{8}-3xyw^{10}-384y^{2}z^{9}w-288y^{2}z^{7}w^{3}-72y^{2}z^{5}w^{5}+144y^{2}z^{3}w^{7}-15y^{2}zw^{9}+80z^{10}w^{2}+52z^{8}w^{4}-56z^{6}w^{6}-8z^{4}w^{8}+2z^{2}w^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
48.72.4.t.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -2X^{6}+2X^{4}YZ-16X^{2}Y^{4}+2X^{2}Y^{2}Z^{2}-X^{2}Z^{4}+6Y^{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.
