Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ x y v - x z v - x t v + w u v + t u v $ |
| $=$ | $x^{2} v - x y v - x z v + x w v + x u v - y u v - w u v - t u v - u^{2} v$ |
| $=$ | $x y v + x z v - x t v + y w v - 2 w u v - t^{2} v$ |
| $=$ | $x y v - x t v + y z v + y w v + y u v - z u v - w t v - t^{2} v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{7} + x^{6} z - 44 x^{5} y^{2} + 9 x^{5} z^{2} + 30 x^{4} y^{2} z - 5 x^{4} z^{3} - 30 x^{3} y^{2} z^{2} + \cdots + 22 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 10x^{11} + 110x^{6} - 10x $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{17}\cdot\frac{1433100000xu^{10}+35855100000xu^{8}v^{2}+286290225000xu^{6}v^{4}-587722211000xu^{4}v^{6}+382534032350xu^{2}v^{8}-94426113307xv^{10}-212500000yt^{10}+361250000yt^{8}v^{2}-128775000yt^{6}v^{4}+24480000yt^{4}v^{6}-1360850yt^{2}v^{8}+146180100000yu^{10}+656360590000yu^{8}v^{2}-1461470307000yu^{6}v^{4}+974978114600yu^{4}v^{6}-251497440780yu^{2}v^{8}+4517500957yv^{10}+212500000zt^{10}+127500000zt^{8}v^{2}+36125000zt^{6}v^{4}-10667500zt^{4}v^{6}-8571400zt^{2}v^{8}+327100400000zu^{10}+179162560000zu^{8}v^{2}-1260842623000zu^{6}v^{4}+1020200551400zu^{4}v^{6}-279549529970zu^{2}v^{8}-4522840657zv^{10}-425000000wt^{10}+425000000wt^{8}v^{2}-7650000wt^{6}v^{4}+2847500wt^{4}v^{6}-14486550wt^{2}v^{8}+115354800000wu^{10}+297273020000wu^{8}v^{2}-949429786000wu^{6}v^{4}+730643436800wu^{4}v^{6}-204419045190wu^{2}v^{8}-1513047574wv^{10}+425000000t^{11}+63750000t^{9}v^{2}-39950000t^{7}v^{4}+28475000t^{5}v^{6}-12037700t^{3}v^{8}-286335500000t^{2}u^{9}+867592700000t^{2}u^{7}v^{2}-877097450000t^{2}u^{5}v^{4}+394210187000t^{2}u^{3}v^{6}-75395954950t^{2}uv^{8}-307442300000tu^{10}-1775605070000tu^{8}v^{2}+3695049991000tu^{6}v^{4}-2312973894300tu^{4}v^{6}+529110809090tu^{2}v^{8}+9037563134tv^{10}+2318800000u^{11}+68672300000u^{9}v^{2}+57372925000u^{7}v^{4}-278726773000u^{5}v^{6}+214650574050u^{3}v^{8}-58514851236uv^{10}}{v^{10}(7x-4y+4z+3w-3t+11u)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.120.5.cx.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{7}-44X^{5}Y^{2}+X^{6}Z+30X^{4}Y^{2}Z+9X^{5}Z^{2}-30X^{3}Y^{2}Z^{2}-5X^{4}Z^{3}-20X^{2}Y^{2}Z^{3}-40X^{3}Z^{4}-22X^{2}Z^{5}-2Y^{2}Z^{5}-7XZ^{6}+22Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.120.5.cx.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{2}{5}x-\frac{1}{5}u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{22}{625}x^{5}v-\frac{3}{125}x^{4}uv+\frac{3}{125}x^{3}u^{2}v+\frac{2}{125}x^{2}u^{3}v+\frac{1}{625}u^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}x+\frac{2}{5}u$ |
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.
