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 $ | $=$ | $ 2 x^{5} + 2 x^{4} z - 4 x^{3} y^{2} + x^{3} z^{2} + 8 x^{2} y^{2} z + x^{2} z^{3} + 6 x y^{4} + \cdots - z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{5} - 2x^{4} - 2x^{2} - x $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^2}\cdot\frac{131113xyz^{18}+3935183xyz^{16}w^{2}-27203610xyz^{14}w^{4}-244727732xyz^{12}w^{6}+498982400xyz^{10}w^{8}+464087376xyz^{8}w^{10}+514229696xyz^{6}w^{12}+77616256xyz^{4}w^{14}-582018816xyz^{2}w^{16}+33197824xyw^{18}+266817xz^{19}+4191683xz^{17}w^{2}-55681158xz^{15}w^{4}-261103060xz^{13}w^{6}+731654816xz^{11}w^{8}+316142768xz^{9}w^{10}+1580558656xz^{7}w^{12}-1232384384xz^{5}w^{14}-553865984xz^{3}w^{16}+194784000xzw^{18}-132620y^{2}z^{18}-349695y^{2}z^{16}w^{2}+35842104y^{2}z^{14}w^{4}+19978948y^{2}z^{12}w^{6}-574542336y^{2}z^{10}w^{8}+434011344y^{2}z^{8}w^{10}-842231296y^{2}z^{6}w^{12}+1919783808y^{2}z^{4}w^{14}-776994816y^{2}z^{2}w^{16}+33195264y^{2}w^{18}+65577yz^{19}+1839550yz^{17}w^{2}-10796025yz^{15}w^{4}-83470300yz^{13}w^{6}+98960664yz^{11}w^{8}-148735232yz^{9}w^{10}+762347104yz^{7}w^{12}-617048448yz^{5}w^{14}+1195522304yz^{3}w^{16}-313163776yzw^{18}+199733z^{20}+2400237z^{18}w^{2}-45331037z^{16}w^{4}-170902352z^{14}w^{6}+637826296z^{12}w^{8}+274970480z^{10}w^{10}+993237728z^{8}w^{12}-901424896z^{6}w^{14}-784903424z^{4}w^{16}+237938432z^{2}w^{18}+2560w^{20}}{w^{8}(xyz^{10}+751xyz^{8}w^{2}-58xyz^{6}w^{4}-2452xyz^{4}w^{6}-1312xyz^{2}w^{8}+432xyw^{10}+1153xz^{11}-5837xz^{9}w^{2}-15854xz^{7}w^{4}-14932xz^{5}w^{6}-8256xz^{3}w^{8}-2160xzw^{10}-384y^{2}z^{10}+1693y^{2}z^{8}w^{2}+5264y^{2}z^{6}w^{4}+6132y^{2}z^{4}w^{6}+3968y^{2}z^{2}w^{8}+432y^{2}w^{10}+yz^{11}+1134yz^{9}w^{2}-1377yz^{7}w^{4}-7084yz^{5}w^{6}-6984yz^{3}w^{8}-1728yzw^{10}+769z^{12}-4143z^{10}w^{2}-10221z^{8}w^{4}-8272z^{6}w^{6}-3688z^{4}w^{8}-816z^{2}w^{10})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
16.96.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{5}-4X^{3}Y^{2}+6XY^{4}+2X^{4}Z+8X^{2}Y^{2}Z-2Y^{4}Z+X^{3}Z^{2}+8XY^{2}Z^{2}+X^{2}Z^{3}-4Y^{2}Z^{3}-XZ^{4}-Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
16.96.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{3}{2}y^{2}z+yz^{2}+yw^{2}-\frac{1}{2}z^{3}-\frac{1}{3}zw^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{4}{3}y^{8}w+\frac{10}{9}y^{7}zw-\frac{275}{27}y^{6}z^{2}w-\frac{8}{3}y^{6}w^{3}-\frac{221}{18}y^{5}z^{3}w-\frac{56}{9}y^{5}zw^{3}+\frac{53}{6}y^{4}z^{4}w+\frac{46}{27}y^{4}z^{2}w^{3}+\frac{55}{9}y^{3}z^{5}w+\frac{16}{3}y^{3}z^{3}w^{3}-\frac{52}{9}y^{2}z^{6}w-\frac{100}{27}y^{2}z^{4}w^{3}+\frac{3}{2}yz^{7}w+\frac{8}{9}yz^{5}w^{3}-\frac{7}{54}z^{8}w-\frac{2}{27}z^{6}w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y^{3}+\frac{7}{6}y^{2}z-yz^{2}+\frac{1}{6}z^{3}$ |
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.
