Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x z t - x w t - y w t - y t^{2} $ |
| $=$ | $2 x z w - x w^{2} - y w^{2} - y w t$ |
| $=$ | $2 x z^{2} - x z w - y z w - y z t$ |
| $=$ | $2 x y z - x y w - y^{2} w - y^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 5 x^{3} y^{2} - 5 x^{3} z^{2} + 10 x^{2} y^{2} z - 6 x^{2} z^{3} - 10 x y^{2} z^{2} + \cdots - 20 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -5x^{7} + 35x^{5} - 35x^{3} + 5x $ |
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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 |
|---|
| $(0:0:-1/2:-1:1)$, $(0:1:0:0:0)$, $(1:0:0:0:0)$, $(-1:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{78125x^{14}+93750x^{12}t^{2}+68750x^{10}t^{4}-15000x^{8}t^{6}+142750x^{6}t^{8}-280800x^{4}t^{10}+624540x^{2}t^{12}+156250xy^{13}+718750xy^{11}t^{2}+175000xy^{9}t^{4}+775000xy^{7}t^{6}+849250xy^{5}t^{8}+2752550xy^{3}t^{10}+6047680xyt^{12}+312500y^{12}t^{2}-50000y^{10}t^{4}+555000y^{8}t^{6}+1048000y^{6}t^{8}+2574700y^{4}t^{10}+6906400y^{2}t^{12}-640zw^{13}+128zw^{12}t-4352zw^{11}t^{2}+256zw^{10}t^{3}-43904zw^{9}t^{4}+4480zw^{8}t^{5}-232960zw^{7}t^{6}+382152zw^{6}t^{7}-1223424zw^{5}t^{8}+2461408zw^{4}t^{9}-3658176zw^{3}t^{10}+5776040zw^{2}t^{11}-4838272zwt^{12}+640zt^{13}+224w^{14}-64w^{13}t+1568w^{12}t^{2}-384w^{11}t^{3}+15456w^{10}t^{4}-4032w^{9}t^{5}+83104w^{8}t^{6}-153428w^{7}t^{7}+430884w^{6}t^{8}-1026920w^{5}t^{9}+1319744w^{4}t^{10}-2783260w^{3}t^{11}+1862948w^{2}t^{12}-1631160wt^{13}+224t^{14}}{t^{4}(125x^{6}t^{4}-250x^{4}t^{6}+590x^{2}t^{8}+250xy^{5}t^{4}-50xy^{3}t^{6}+2960xyt^{8}+500y^{4}t^{6}+1360y^{2}t^{8}-40zw^{9}+8zw^{8}t-272zw^{7}t^{2}+16zw^{6}t^{3}-904zw^{5}t^{4}-88zw^{4}t^{5}-2048zw^{3}t^{6}-184zw^{2}t^{7}-2368zwt^{8}+14w^{10}-4w^{9}t+98w^{8}t^{2}-24w^{7}t^{3}+322w^{6}t^{4}-68w^{5}t^{5}+686w^{4}t^{6}-236w^{3}t^{7}+732w^{2}t^{8}-508wt^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.96.3.be.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{3}Y^{2}-X^{4}Z+10X^{2}Y^{2}Z-5X^{3}Z^{2}-10XY^{2}Z^{2}-6X^{2}Z^{3}-20Y^{2}Z^{3}-2XZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.96.3.be.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x-y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x^{3}t+2x^{2}yt-2xy^{2}t-4y^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
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.
