Embedded model Embedded model in $\mathbb{P}^{6}$
$ 0 $ | $=$ | $ x y v + t u v $ |
| $=$ | $x y v - z v^{2}$ |
| $=$ | $x^{2} v + t^{2} u$ |
| $=$ | $x^{2} v - z t v$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{11} + x y^{2} z^{8} - 2 y z^{10} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{12} - 1 $ |
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 embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{13ywuv^{4}+48zw^{4}uv-76zu^{3}v^{3}-4zv^{6}-8w^{7}+96w^{3}u^{2}v^{2}}{vuw^{4}z}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.72.5.d.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}v$ |
Equation of the image curve:
$0$ |
$=$ |
$ X^{11}+XY^{2}Z^{8}-2YZ^{10} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.72.5.d.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}v$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{4}wtv^{4}+\frac{1}{64}v^{6}$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle -t$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.