Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z w - x w^{2} - y w^{2} $ |
| $=$ | $x z^{2} - x z w - y z w$ |
| $=$ | $x^{2} z - x^{2} w - x y w$ |
| $=$ | $x y z - x y w - y^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + x^{2} y^{2} + 2 x y^{2} z - y^{2} z^{2} - 2 z^{4} $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^2}\cdot\frac{4194304x^{20}+18612224x^{18}w^{2}+153550848x^{16}w^{4}-3029467136x^{14}w^{6}+9634119680x^{12}w^{8}-7687176192x^{10}w^{10}+12993454080x^{8}w^{12}-74483589120x^{6}w^{14}+98156113920x^{4}w^{16}-82307460096x^{2}w^{18}+9062285312xy^{19}+258840377344xy^{17}w^{2}+1879693974528xy^{15}w^{4}+2166864538624xy^{13}w^{6}+790020834304xy^{11}w^{8}+1717003129856xy^{9}w^{10}-4306471447552xy^{7}w^{12}+15969875772416xy^{5}w^{14}-3257632887808xy^{3}w^{16}+26469268444160xyw^{18}+2782294016y^{20}+82173890048y^{18}w^{2}+654062638336y^{16}w^{4}+1048285360128y^{14}w^{6}-29780075520y^{12}w^{8}+2238915082240y^{10}w^{10}-7824608877056y^{8}w^{12}+24760263546880y^{6}w^{14}+1738812156928y^{4}w^{16}+45036817906176y^{2}w^{18}-2684936z^{20}+45102576z^{19}w-293611337z^{18}w^{2}+1114344046z^{17}w^{3}-3444035294z^{16}w^{4}+9078099456z^{15}w^{5}-19848799172z^{14}w^{6}+37819244104z^{13}w^{7}-58632663880z^{12}w^{8}+67773565792z^{11}w^{9}-25196778462z^{10}w^{10}-38694825292z^{9}w^{11}+172250458284z^{8}w^{12}+193349359616z^{7}w^{13}-172626939092z^{6}w^{14}+1342516549384z^{5}w^{15}-2744325801888z^{4}w^{16}+6395443813808z^{3}w^{17}+6885997838911z^{2}w^{18}+5327373775502zw^{19}+9301944217506w^{20}}{w^{4}(65536x^{12}w^{4}+552960x^{10}w^{6}+4676608x^{8}w^{8}-28272640x^{6}w^{10}+37421056x^{4}w^{12}-37931520x^{2}w^{14}+26361856xy^{11}w^{4}+1689088xy^{9}w^{6}+1680942848xy^{7}w^{8}-15044860672xy^{5}w^{10}+204190337792xy^{3}w^{12}-2839852403456xyw^{14}+24708096y^{12}w^{4}-139494528y^{10}w^{6}+2192699712y^{8}w^{8}-25354082304y^{6}w^{10}+340817804544y^{4}w^{12}-4747312294144y^{2}w^{14}+8037z^{16}-97304z^{15}w+700706z^{14}w^{2}-4083640z^{13}w^{3}+20164503z^{12}w^{4}-89089344z^{11}w^{5}+358697912z^{10}w^{6}-1345206952z^{9}w^{7}+4638655483z^{8}w^{8}-15239383976z^{7}w^{9}+44805568362z^{6}w^{10}-129266575768z^{5}w^{11}+274612652657z^{4}w^{12}-694643915520z^{3}w^{13}-755137453124z^{2}w^{14}-579356619784zw^{15}-989243966824w^{16})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
16.96.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{4}+X^{2}Y^{2}+2XY^{2}Z-Y^{2}Z^{2}-2Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
16.96.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}z+\frac{1}{2}w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{2}yz^{2}-yzw+\frac{1}{2}yw^{2}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{2}z+\frac{1}{2}w$ |
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.
