Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x^{2} + 24 y^{2} - z^{2} - w^{2} $ |
| $=$ | $9 x^{2} y - 2 x z w + y z^{2} + y w^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} - 3 x^{4} z^{2} + 4 x^{2} y^{4} - 24 x^{2} y^{2} z^{2} + 36 x^{2} z^{4} + 4 y^{6} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\cdot3^3\,\frac{96xyz^{9}w-468xyz^{7}w^{3}+828xyz^{5}w^{5}-468xyz^{3}w^{7}+96xyzw^{9}+156y^{2}z^{10}-612y^{2}z^{8}w^{2}+540y^{2}z^{6}w^{4}+540y^{2}z^{4}w^{6}-612y^{2}z^{2}w^{8}+156y^{2}w^{10}-9z^{12}+25z^{10}w^{2}+2z^{8}w^{4}-55z^{6}w^{6}+2z^{4}w^{8}+25z^{2}w^{10}-9w^{12}}{60xyz^{9}w+72xyz^{7}w^{3}+72xyz^{5}w^{5}+72xyz^{3}w^{7}+60xyzw^{9}-24y^{2}z^{10}+144y^{2}z^{8}w^{2}+216y^{2}z^{6}w^{4}+216y^{2}z^{4}w^{6}+144y^{2}z^{2}w^{8}-24y^{2}w^{10}-8z^{10}w^{2}-19z^{8}w^{4}-22z^{6}w^{6}-19z^{4}w^{8}-8z^{2}w^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
12.72.4.j.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y^{2}-3X^{4}Z^{2}+4X^{2}Y^{4}-24X^{2}Y^{2}Z^{2}+36X^{2}Z^{4}+4Y^{6} $ |
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.
