Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
$ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $2 x y + 5 x z + 2 y^{2} + 3 z^{2} + w^{2}$ |
| $=$ | $5 x^{2} + 3 x y - 10 x z + 3 y^{2} + 7 z^{2} + 2 w^{2} + 2 t^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 486875 x^{8} - 11500 x^{7} y + 2400 x^{6} y^{2} + 222500 x^{6} z^{2} - 40 x^{5} y^{3} - 5300 x^{5} y z^{2} + \cdots + 35 z^{8} $ |
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^6\,\frac{156240z^{2}w^{16}+350640z^{2}w^{14}t^{2}+258480z^{2}w^{12}t^{4}-258480z^{2}w^{10}t^{6}-1213200z^{2}w^{8}t^{8}-1735920z^{2}w^{6}t^{10}-1151820z^{2}w^{4}t^{12}-351540z^{2}w^{2}t^{14}-39060z^{2}t^{16}+6248w^{18}+10752w^{16}t^{2}+432w^{14}t^{4}-74860w^{12}t^{6}-282720w^{10}t^{8}-466944w^{8}t^{10}-392186w^{6}t^{12}-172656w^{4}t^{14}-37500w^{2}t^{16}-3125t^{18}}{t^{2}w^{4}(80z^{2}w^{10}-100z^{2}w^{8}t^{2}+100z^{2}w^{6}t^{4}-100z^{2}w^{4}t^{6}-200z^{2}w^{2}t^{8}-40z^{2}t^{10}+16w^{12}-12w^{10}t^{2}+9w^{8}t^{4}-8w^{6}t^{6}-4w^{4}t^{8})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.5.iy.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 5y+5t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
$0$ |
$=$ |
$ 486875X^{8}-11500X^{7}Y+2400X^{6}Y^{2}-40X^{5}Y^{3}+4X^{4}Y^{4}+222500X^{6}Z^{2}-5300X^{5}YZ^{2}+640X^{4}Y^{2}Z^{2}-8X^{3}Y^{3}Z^{2}+33750X^{4}Z^{4}-700X^{3}YZ^{4}+24X^{2}Y^{2}Z^{4}+1900X^{2}Z^{6}-20XYZ^{6}+35Z^{8} $ |
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.