Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x z w - y z t $ |
| $=$ | $x y w - y^{2} t$ |
| $=$ | $x w t - y t^{2}$ |
| $=$ | $x w^{2} - y w t$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} y + 5 x^{4} z - 3 x^{2} y^{2} z + 6 x^{2} y z^{2} + 6 x^{2} z^{3} + 9 y z^{4} + 9 z^{5} $ |
Weierstrass model Weierstrass model
$ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 6x^{6} + 49x^{4} + 270x^{2} + 506 $ |
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 |
$(0:0:0:0:1)$, $(1:0:0:0:0)$, $(1:-1:1:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{474609375x^{11}+506250000x^{9}t^{2}+141328125x^{7}t^{4}-16875000x^{5}t^{6}-8831250x^{3}t^{8}+1405000xt^{10}+474609375yz^{10}+348046875yz^{8}t^{2}+154828125yz^{6}t^{4}+27613125yz^{4}t^{6}+1527750yz^{2}t^{8}-3089408yw^{10}-19243008yw^{9}t-83523584yw^{8}t^{2}-197888768yw^{7}t^{3}-301738048yw^{6}t^{4}-295364864yw^{5}t^{5}-173919712yw^{4}t^{6}-33864400yw^{3}t^{7}+33598200yw^{2}t^{8}+32407312ywt^{9}+12433734yt^{10}+38812500z^{7}t^{4}+8010000z^{5}t^{6}+3810000z^{3}t^{8}-8385536zw^{10}-42356736zw^{9}t-132161536zw^{8}t^{2}-252640256zw^{7}t^{3}-309848768zw^{6}t^{4}-229691904zw^{5}t^{5}-64871168zw^{4}t^{6}+52239904zw^{3}t^{7}+69537720zw^{2}t^{8}+33348928zwt^{9}+7024984zt^{10}}{5625yz^{4}t^{6}+1792yw^{10}+11136yw^{9}t+25984yw^{8}t^{2}+31648yw^{7}t^{3}+22560yw^{6}t^{4}+9672yw^{5}t^{5}+2488yw^{4}t^{6}+310yw^{3}t^{7}-45yw^{2}t^{8}-20ywt^{9}+5625z^{5}t^{6}-375z^{3}t^{8}+4864zw^{10}+24512zw^{9}t+49920zw^{8}t^{2}+54160zw^{7}t^{3}+34192zw^{6}t^{4}+12724zw^{5}t^{5}+2728zw^{4}t^{6}+295zw^{3}t^{7}-50zw^{2}t^{8}+5zwt^{9}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.ca.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
$0$ |
$=$ |
$ 5X^{4}Y+5X^{4}Z-3X^{2}Y^{2}Z+6X^{2}YZ^{2}+6X^{2}Z^{3}+9YZ^{4}+9Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.ca.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -w$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 22y^{4}+3y^{2}w^{2}-3y^{2}wt$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
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.