Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} w + y^{2} w - z^{2} w $ |
| $=$ | $2 x^{2} z + y^{2} z - z^{3}$ |
| $=$ | $2 x^{2} y + y^{3} - y z^{2}$ |
| $=$ | $2 x^{3} + x y^{2} - x z^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{5} + 4 x^{4} z + 4 x^{3} y^{2} + 2 x^{3} z^{2} - 8 x^{2} y^{2} z + 2 x^{2} z^{3} + 3 x y^{4} + \cdots - 2 z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{5} + 4x^{4} + 4x^{2} - 2x $ |
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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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{524452xyz^{18}-7870366xyz^{16}w^{2}-27203610xyz^{14}w^{4}+122363866xyz^{12}w^{6}+124745600xyz^{10}w^{8}-58010922xyz^{8}w^{10}+32139356xyz^{6}w^{12}-2425508xyz^{4}w^{14}-9094044xyz^{2}w^{16}-259358xyw^{18}+1067268xz^{19}-8383366xz^{17}w^{2}-55681158xz^{15}w^{4}+130551530xz^{13}w^{6}+182913704xz^{11}w^{8}-39517846xz^{9}w^{10}+98784916xz^{7}w^{12}+38512012xz^{5}w^{14}-8654156xz^{3}w^{16}-1521750xzw^{18}+530480y^{2}z^{18}-699390y^{2}z^{16}w^{2}-35842104y^{2}z^{14}w^{4}+9989474y^{2}z^{12}w^{6}+143635584y^{2}z^{10}w^{8}+54251418y^{2}z^{8}w^{10}+52639456y^{2}z^{6}w^{12}+59993244y^{2}z^{4}w^{14}+12140544y^{2}z^{2}w^{16}+259338y^{2}w^{18}-262308yz^{19}+3679100yz^{17}w^{2}+10796025yz^{15}w^{4}-41735150yz^{13}w^{6}-24740166yz^{11}w^{8}-18591904yz^{9}w^{10}-47646694yz^{7}w^{12}-19282764yz^{5}w^{14}-18680036yz^{3}w^{16}-2446592yzw^{18}-798932z^{20}+4800474z^{18}w^{2}+45331037z^{16}w^{4}-85451176z^{14}w^{6}-159456574z^{12}w^{8}+34371310z^{10}w^{10}-62077358z^{8}w^{12}-28169528z^{6}w^{14}+12264116z^{4}w^{16}+1858894z^{2}w^{18}-10w^{20}}{w^{8}(4xyz^{10}-1502xyz^{8}w^{2}-58xyz^{6}w^{4}+1226xyz^{4}w^{6}-328xyz^{2}w^{8}-54xyw^{10}+4612xz^{11}+11674xz^{9}w^{2}-15854xz^{7}w^{4}+7466xz^{5}w^{6}-2064xz^{3}w^{8}+270xzw^{10}+1536y^{2}z^{10}+3386y^{2}z^{8}w^{2}-5264y^{2}z^{6}w^{4}+3066y^{2}z^{4}w^{6}-992y^{2}z^{2}w^{8}+54y^{2}w^{10}-4yz^{11}+2268yz^{9}w^{2}+1377yz^{7}w^{4}-3542yz^{5}w^{6}+1746yz^{3}w^{8}-216yzw^{10}-3076z^{12}-8286z^{10}w^{2}+10221z^{8}w^{4}-4136z^{6}w^{6}+922z^{4}w^{8}-102z^{2}w^{10})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
16.96.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{5}+4X^{3}Y^{2}+3XY^{4}+4X^{4}Z-8X^{2}Y^{2}Z-Y^{4}Z+2X^{3}Z^{2}-8XY^{2}Z^{2}+2X^{2}Z^{3}+4Y^{2}Z^{3}-2XZ^{4}-2Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
16.96.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{3}{2}y^{2}z-yz^{2}+\frac{1}{2}yw^{2}+\frac{1}{2}z^{3}-\frac{1}{6}zw^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{4}{3}y^{8}w+\frac{10}{9}y^{7}zw-\frac{275}{27}y^{6}z^{2}w+\frac{4}{3}y^{6}w^{3}-\frac{221}{18}y^{5}z^{3}w+\frac{28}{9}y^{5}zw^{3}+\frac{53}{6}y^{4}z^{4}w-\frac{23}{27}y^{4}z^{2}w^{3}+\frac{55}{9}y^{3}z^{5}w-\frac{8}{3}y^{3}z^{3}w^{3}-\frac{52}{9}y^{2}z^{6}w+\frac{50}{27}y^{2}z^{4}w^{3}+\frac{3}{2}yz^{7}w-\frac{4}{9}yz^{5}w^{3}-\frac{7}{54}z^{8}w+\frac{1}{27}z^{6}w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y^{3}+\frac{7}{6}y^{2}z-yz^{2}+\frac{1}{6}z^{3}$ |
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.
