Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ - x z t + y z w $ |
| $=$ | $ - x y t + y^{2} w$ |
| $=$ | $ - x t^{2} + y w t$ |
| $=$ | $ - x w t + y w^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} y - x^{4} z - 3 x^{2} y^{2} z - 6 x^{2} y z^{2} + 6 x^{2} z^{3} + 45 y z^{4} - 45 z^{5} $ |
Weierstrass model Weierstrass model
$ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 6x^{8} - 30x^{6} + 49x^{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 |
$(0:0:0:1:0)$, $(1:0:0:0:0)$, $(1:1:1:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{151875x^{11}-759375x^{9}w^{2}+2784375x^{9}wt-1518750x^{9}t^{2}-13584375x^{7}w^{2}t^{2}+44836875x^{7}wt^{3}-62151057x^{7}t^{4}-10813851x^{5}w^{2}t^{4}+90359775x^{5}wt^{5}-11020266x^{5}t^{6}-110571375x^{3}w^{2}t^{6}+180980643x^{3}wt^{7}+10595772x^{3}t^{8}+28296573xw^{2}t^{8}-66017343xwt^{9}+3096576xt^{10}+1518750yz^{10}-7036875yz^{8}t^{2}+22184307yz^{6}t^{4}+28761120yz^{4}t^{6}-89162721yz^{2}t^{8}-133170044yt^{10}+759375z^{11}+2480625z^{9}t^{2}+1525500z^{7}t^{4}+12449844z^{5}t^{6}+10268217z^{3}t^{8}+16zw^{10}-176zw^{9}t+12400zw^{8}t^{2}-152528zw^{7}t^{3}-25384331zw^{6}t^{4}+47006772zw^{5}t^{5}-136407182zw^{4}t^{6}-100447405zw^{3}t^{7}+49590839zw^{2}t^{8}-8404992zwt^{9}+144695960zt^{10}}{t^{2}(144x^{5}wt^{3}-2880x^{5}t^{4}+7773x^{3}w^{2}t^{4}-15897x^{3}wt^{5}+8121x^{3}t^{6}-7333xw^{2}t^{6}+10530xwt^{7}-1792xt^{8}+3375yz^{6}t^{2}+5661yz^{4}t^{4}+7146yz^{2}t^{6}+9560yt^{8}-3375z^{7}t^{2}-4050z^{5}t^{4}+417z^{3}t^{6}-16zw^{8}+208zw^{7}t-1056zw^{6}t^{2}+2080zw^{5}t^{3}+4144zw^{4}t^{4}+18280zw^{3}t^{5}-18105zw^{2}t^{6}+4864zwt^{7}-8236zt^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.cb.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
$0$ |
$=$ |
$ X^{4}Y-X^{4}Z-3X^{2}Y^{2}Z-6X^{2}YZ^{2}+6X^{2}Z^{3}+45YZ^{4}-45Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.cb.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -t$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4y^{4}-3y^{2}wt-3y^{2}t^{2}+2t^{4}$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
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.