Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z^{2} + x z w - y z^{2} + y z w $ |
| $=$ | $x z w + x w^{2} - y z w + y w^{2}$ |
| $=$ | $x^{2} z + x^{2} w - x y z + x y w$ |
| $=$ | $x y z + x y w - y^{2} z + y^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + 2 x^{4} z + 2 x^{2} y^{2} z - 2 x^{2} z^{3} - x z^{4} + 2 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{5} - x $ |
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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.
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{32767x^{2}y^{8}+2024x^{2}y^{6}w^{2}+636504x^{2}y^{4}w^{4}+86000x^{2}y^{2}w^{6}-2032x^{2}w^{8}-32768xy^{9}+524888xy^{7}w^{2}+2746920xy^{5}w^{4}-427984xy^{3}w^{6}-10272xyw^{8}-y^{10}-131100y^{8}z^{2}+130480y^{8}zw-260148y^{8}w^{2}-1638840y^{6}z^{2}w^{2}-139240y^{6}zw^{3}+1616480y^{6}w^{4}-1597728y^{4}z^{2}w^{4}+2073648y^{4}zw^{5}-475920y^{4}w^{6}+65568y^{2}z^{2}w^{6}-65568y^{2}zw^{7}-12304y^{2}w^{8}+8128z^{2}w^{8}+8192zw^{9}+64w^{10}}{wy^{2}(3x^{2}y^{4}w+111x^{2}y^{2}w^{3}+128x^{2}w^{5}+xy^{5}w+130xy^{3}w^{3}+512xyw^{5}-y^{6}z+3y^{6}w+46y^{4}z^{2}w-150y^{4}zw^{2}+153y^{4}w^{3}+516y^{2}z^{2}w^{3}-960y^{2}zw^{4}+444y^{2}w^{5}+128z^{2}w^{5}-128zw^{6})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
16.48.2.k.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}+2X^{4}Z+2X^{2}Y^{2}Z-2X^{2}Z^{3}+2Y^{2}Z^{3}-XZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
16.48.2.k.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}z^{2}+zw+\frac{1}{2}w^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{4}yz^{4}w-\frac{1}{2}yz^{3}w^{2}-\frac{1}{2}yz^{2}w^{3}-\frac{1}{2}yzw^{4}-\frac{1}{4}yw^{5}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{2}z^{2}+\frac{1}{2}w^{2}$ |
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.
