Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - y u + z t $ |
| $=$ | $x u + z w$ |
| $=$ | $x t + y w$ |
| $=$ | $9 w^{2} + 6 w t - 3 t^{2} + u^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} y^{4} + 10 x^{4} y^{2} z^{2} + 5 x^{4} z^{4} + 6 x^{2} y^{4} z^{2} - 6 x^{2} y^{2} z^{4} + \cdots + 9 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{8} + 4x^{7} - 52x^{6} + 220x^{5} - 748x^{4} + 220x^{3} - 52x^{2} + 4x + 2 $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^6}{3^4}\cdot\frac{81181440xz^{9}-68304384xz^{7}u^{2}-484825824xz^{5}u^{4}-309298176xz^{3}u^{6}+8239401xzu^{8}+33592320yz^{9}+44229888yz^{7}u^{2}-78662016yz^{5}u^{4}-87816960yz^{3}u^{6}-6380532yzu^{8}-196680000wt^{8}u-491016000wt^{6}u^{3}-406630500wt^{4}u^{5}-113959950wt^{2}u^{7}-16213668wu^{9}+65760000t^{9}u+146482000t^{7}u^{3}+94673000t^{5}u^{5}+1054025t^{3}u^{7}-9007675tu^{9}}{37120xz^{9}+17728xz^{7}u^{2}-1008xz^{5}u^{4}-1484xz^{3}u^{6}+177xzu^{8}+15360yz^{9}-896yz^{7}u^{2}-576yz^{5}u^{4}-184yz^{3}u^{6}+68yzu^{8}-648000wt^{6}u^{3}+307800wt^{4}u^{5}-41850wt^{2}u^{7}+1132wu^{9}+216000t^{7}u^{3}-156600t^{5}u^{5}+36225t^{3}u^{7}-2675tu^{9}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.hs.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{4}Y^{4}+10X^{4}Y^{2}Z^{2}+6X^{2}Y^{4}Z^{2}+5X^{4}Z^{4}-6X^{2}Y^{2}Z^{4}+9Y^{4}Z^{4}-12X^{2}Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.hs.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{30}wu+\frac{1}{180}u^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{11}{303750}zwt^{3}u^{3}-\frac{11}{303750}zwt^{2}u^{4}+\frac{11}{1215000}zwtu^{5}-\frac{2}{151875}zt^{4}u^{3}+\frac{2}{151875}zt^{3}u^{4}+\frac{1}{3645000}zt^{2}u^{5}-\frac{13}{3645000}ztu^{6}+\frac{13}{14580000}zu^{7}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{60}wu-\frac{1}{60}tu+\frac{1}{90}u^{2}$ |
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.
