Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x y w - x y t - y^{2} w + y z w + w^{3} $ |
| $=$ | $x y w + 2 x y t - y^{2} w + y z w + w^{3} + w^{2} t$ |
| $=$ | $2 x y w + x y t + y^{2} w - y z w$ |
| $=$ | $3 x y z + z w^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} y + 4 x^{4} z - 3 x^{2} y^{2} z - 12 x^{2} y z^{2} - 15 x^{2} z^{3} + 9 y z^{4} + 36 z^{5} $ |
Weierstrass model Weierstrass model
$ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 6x^{6} - 5x^{4} + 54x^{2} + 20 $ |
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Embedded model |
$(0:0:0:0:1)$, $(0:0:1:0:0)$, $(0:1:1:0:0)$, $(-1/2:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2}\cdot\frac{407558946816xz^{10}+305713004544xz^{8}t^{2}-26163689472xz^{6}t^{4}-231011136576xz^{4}t^{6}-676002211626xz^{2}t^{8}-3971447708xt^{10}-204862390272yz^{10}-194694119424yz^{8}t^{2}-10236284928yz^{6}t^{4}+119258296488yz^{4}t^{6}+360584539353yz^{2}t^{8}-17129878517760yw^{10}-27400266185728yw^{9}t+27694542697856yw^{8}t^{2}+83921870872064yw^{7}t^{3}+45630192510376yw^{6}t^{4}-33028680671924yw^{5}t^{5}-60985791011802yw^{4}t^{6}-39770355462912yw^{3}t^{7}-12954615914530yw^{2}t^{8}-1814799556016ywt^{9}+1677721600yt^{10}+203843174400z^{11}+195392176128z^{9}t^{2}+8425783296z^{7}t^{4}-119714715648z^{5}t^{6}-360504998208z^{3}t^{8}+1362870120448zw^{10}-16546947734528zw^{9}t-31467088636160zw^{8}t^{2}+23734136133696zw^{7}t^{3}+82535186182128zw^{6}t^{4}+43715663371140zw^{5}t^{5}-31113514820082zw^{4}t^{6}-52210715911103zw^{3}t^{7}-30344372768992zw^{2}t^{8}-8290652647973zwt^{9}-912963801354zt^{10}}{442368xz^{6}t^{4}-354816xz^{4}t^{6}+26304xz^{2}t^{8}-578789xt^{10}+221184yz^{6}t^{4}+46080yz^{4}t^{6}-33732yz^{2}t^{8}+5772902400yw^{10}+1210982400yw^{9}t-3323289600yw^{8}t^{2}-2916633600yw^{7}t^{3}-720295680yw^{6}t^{4}-29527104yw^{5}t^{5}+9333392yw^{4}t^{6}-3471972yw^{3}t^{7}+1023474yw^{2}t^{8}-690533ywt^{9}-147456z^{5}t^{6}+44544z^{3}t^{8}-398131200zw^{10}+6104678400zw^{9}t+611020800zw^{8}t^{2}-3737548800zw^{7}t^{3}-2189229056zw^{6}t^{4}-382860288zw^{5}t^{5}-9504000zw^{4}t^{6}+3723776zw^{3}t^{7}-1780834zw^{2}t^{8}+399993zwt^{9}-55872zt^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.a.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
$0$ |
$=$ |
$ X^{4}Y+4X^{4}Z-3X^{2}Y^{2}Z-12X^{2}YZ^{2}-15X^{2}Z^{3}+9YZ^{4}+36Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.a.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -w$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4y^{4}-6y^{2}w^{2}-3y^{2}wt$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
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.