Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z w + x z t + y z t $ |
| $=$ | $x w^{2} + x w t + y w t$ |
| $=$ | $x w t + x t^{2} + y t^{2}$ |
| $=$ | $x^{2} w + x^{2} t + x y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} y^{2} + 5 x^{4} y z - 6 x^{2} y^{2} z^{2} - 6 x^{2} y z^{3} + 3 x^{2} z^{4} + 9 y^{2} z^{4} + 9 y z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -6x^{6} + 49x^{4} - 270x^{2} + 506 $ |
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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 |
|---|
| $(0:0:0:-1:1)$, $(1:0:0:0:0)$, $(-1:1:1:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{474609375x^{11}-506250000x^{9}t^{2}+141328125x^{7}t^{4}+16875000x^{5}t^{6}-8831250x^{3}t^{8}-1405000xt^{10}+474609375yz^{10}-348046875yz^{8}t^{2}+154828125yz^{6}t^{4}-27613125yz^{4}t^{6}+1527750yz^{2}t^{8}+3089408yw^{10}+11651072yw^{9}t+49359872yw^{8}t^{2}+148280576yw^{7}t^{3}+287540032yw^{6}t^{4}+390631808yw^{5}t^{5}+367866784yw^{4}t^{6}+248456880yw^{3}t^{7}+110020248yw^{2}t^{8}+33574832ywt^{9}+2285090yt^{10}-38812500z^{7}t^{4}+8010000z^{5}t^{6}-3810000z^{3}t^{8}-8385536zw^{10}-41498624zw^{9}t-128300032zw^{8}t^{2}-286073856zw^{7}t^{3}-444886720zw^{6}t^{4}-501207680zw^{5}t^{5}-397055552zw^{4}t^{6}-218716448zw^{3}t^{7}-74805736zw^{2}t^{8}-12566192zwt^{9}+395760zt^{10}}{5625yz^{4}t^{6}+1792yw^{10}+6784yw^{9}t+6400yw^{8}t^{2}-9632yw^{7}t^{3}-30528yw^{6}t^{4}-35368yw^{5}t^{5}-23088yw^{4}t^{6}-8838yw^{3}t^{7}-1679yw^{2}t^{8}-16ywt^{9}+33yt^{10}-5625z^{5}t^{6}-375z^{3}t^{8}-4864zw^{10}-24128zw^{9}t-48192zw^{8}t^{2}-46448zw^{7}t^{3}-15264zw^{6}t^{4}+12196zw^{5}t^{5}+16284zw^{4}t^{6}+8191zw^{3}t^{7}+2079zw^{2}t^{8}+274zwt^{9}+42zt^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.cb.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{4}Y^{2}+5X^{4}YZ-6X^{2}Y^{2}Z^{2}-6X^{2}YZ^{3}+3X^{2}Z^{4}+9Y^{2}Z^{4}+9YZ^{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.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -45x^{4}wt^{3}-23x^{4}t^{4}+6x^{2}wt^{5}+3x^{2}t^{6}-wt^{7}-t^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle xt$ |
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.
