Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + y^{2} + 2 y z + z^{2} + w^{2} $ |
| $=$ | $3 x y - 3 x z - 2 y^{2} + 2 y z - 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} - 8 x^{3} z + 3 x^{2} y^{2} + 6 x^{2} z^{2} - 6 x y^{2} z - 8 x z^{3} + 3 y^{2} z^{2} + 7 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^2}{3^2}\cdot\frac{12474315715968xz^{11}+34995621532032xz^{9}w^{2}+27173923737408xz^{7}w^{4}+7528417150464xz^{5}w^{6}+618796879848xz^{3}w^{8}+8024849169408y^{2}z^{10}+7943171647824y^{2}z^{8}w^{2}-1427076239904y^{2}z^{6}w^{4}-2778046125120y^{2}z^{4}w^{6}-715451862384y^{2}z^{2}w^{8}-41250882309y^{2}w^{10}+21966857856yz^{11}-9888429100896yz^{9}w^{2}-13899322785600yz^{7}w^{4}-5937124463424yz^{5}w^{6}-806736798504yz^{3}w^{8}-41981777922yzw^{10}+13188843258048z^{12}+25474813239312z^{10}w^{2}+11235138013008z^{8}w^{4}-2112596896032z^{6}w^{6}-2097532768212z^{4}w^{8}-350649322233z^{2}w^{10}-18078415936w^{12}}{3208414536xz^{11}+2222196984xz^{9}w^{2}+114762312xz^{7}w^{4}-6947808xz^{5}w^{6}+576240xz^{3}w^{8}+2064004416y^{2}z^{10}+71221437y^{2}z^{8}w^{2}-83161722y^{2}z^{6}w^{4}+5356631y^{2}z^{4}w^{6}-657874y^{2}z^{2}w^{8}+50421y^{2}w^{10}+5649912yz^{11}-1002348054yz^{9}w^{2}-153011544yz^{7}w^{4}+14211862yz^{5}w^{6}-102900yz^{3}w^{8}+187278yzw^{10}+3392192196z^{12}+1716929649z^{10}w^{2}-115872330z^{8}w^{4}-28176421z^{6}w^{6}+1212162z^{4}w^{8}-237699z^{2}w^{10}}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.