Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t + y^{2} $ |
| $=$ | $y v + 2 w t + t u$ |
| $=$ | $ - x v + 2 y w + y u$ |
| $=$ | $2 y w - y u + z t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 625 x^{8} + 500 x^{6} y^{2} + 125 x^{6} z^{2} + 150 x^{4} y^{4} - 25 x^{4} y^{2} z^{2} + 20 x^{2} y^{6} + \cdots + y^{6} z^{2} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=31$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^6\,\frac{15625z^{12}-187500z^{11}v+656250z^{10}v^{2}-1187500z^{9}v^{3}+2109375z^{8}v^{4}-4125000z^{7}v^{5}+30183413z^{6}v^{6}-123220900z^{5}v^{7}+145053880z^{4}v^{8}-33904484z^{3}v^{9}+293260029z^{2}v^{10}-4086zu^{10}v+1083258zu^{8}v^{3}-160392412zu^{6}v^{5}-14547614zu^{4}v^{7}+651275562zu^{2}v^{9}+86192604zv^{11}+64000000wu^{11}+320025284wu^{9}v^{2}+422307088wu^{7}v^{4}-873448316wu^{5}v^{6}-2225742704wu^{3}v^{8}-815911144wuv^{10}-80019530t^{2}u^{10}-320021615t^{2}u^{8}v^{2}-207836145t^{2}u^{6}v^{4}+1462675730t^{2}u^{4}v^{6}+1523139895t^{2}u^{2}v^{8}+442309925t^{2}v^{10}-31999999u^{12}-144016548u^{10}v^{2}-114057644u^{8}v^{4}+495278393u^{6}v^{6}+1053151493u^{4}v^{8}+744668535u^{2}v^{10}+88461986v^{12}}{v(16z^{6}v^{5}+556z^{5}v^{6}-3147z^{4}v^{7}+10560z^{3}v^{8}-8936z^{2}v^{9}+16zu^{10}-748zu^{8}v^{2}-1709zu^{6}v^{4}-5202zu^{4}v^{6}+7739zu^{2}v^{8}+9960zv^{10}+384wu^{9}v-3660wu^{7}v^{3}-10276wu^{5}v^{5}-32264wu^{3}v^{7}-37352wuv^{9}-500t^{2}u^{8}v+5205t^{2}u^{6}v^{3}+5750t^{2}u^{4}v^{5}+30385t^{2}u^{2}v^{7}-14725t^{2}v^{9}-192u^{10}v+1274u^{8}v^{3}+1819u^{6}v^{5}+4781u^{4}v^{7}+4257u^{2}v^{9}-2945v^{11})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fb.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 4w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 625X^{8}+500X^{6}Y^{2}+125X^{6}Z^{2}+150X^{4}Y^{4}-25X^{4}Y^{2}Z^{2}+20X^{2}Y^{6}-5X^{2}Y^{4}Z^{2}+5X^{2}Y^{2}Z^{4}+Y^{8}+Y^{6}Z^{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.
