Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 x^{2} - x y - w t - w u - t^{2} - u^{2} $ |
| $=$ | $x y - 2 y^{2} - w^{2} - w t - w u - t^{2} - u^{2}$ |
| $=$ | $2 x^{2} + 2 x y - t^{2} + 2 t u - u^{2}$ |
| $=$ | $x w + x t + x u - 2 z t + 2 z u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{6} y^{2} + 4 x^{6} z^{2} + 8 x^{5} y^{2} z - 4 x^{5} z^{3} + 4 x^{4} y^{4} + 4 x^{4} y^{2} z^{2} + \cdots + 4 z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -6x^{8} - 80x^{4} - 96 $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^3\cdot3^3\,\frac{98171xt^{8}-597432xt^{7}u+2256116xt^{6}u^{2}-6442248xt^{5}u^{3}+14711106xt^{4}u^{4}-27097224xt^{3}u^{5}+36936884xt^{2}u^{6}-21252408xtu^{7}+24383771xu^{8}+192yw^{8}-3072yw^{6}u^{2}-12288yw^{5}u^{3}-67584yw^{4}u^{4}-491520yw^{3}u^{5}-3843072yw^{2}u^{6}-31862784ywu^{7}-152732yt^{8}+1080720yt^{7}u-4311616yt^{6}u^{2}+12656784yt^{5}u^{3}-29493720yt^{4}u^{4}+55403568yt^{3}u^{5}-79417120yt^{2}u^{6}+54129072ytu^{7}-38347820yu^{8}-322496zt^{8}+1837440zt^{7}u-6229824zt^{6}u^{2}+15752064zt^{5}u^{3}-30969024zt^{4}u^{4}+45637248zt^{3}u^{5}-37015488zt^{2}u^{6}-32297856ztu^{7}+43607936zu^{8}}{5xt^{8}+6xt^{7}u-4xt^{6}u^{2}+18xt^{5}u^{3}-18xt^{4}u^{4}+18xt^{3}u^{5}-4xt^{2}u^{6}+6xtu^{7}+5xu^{8}-8yt^{8}+8yt^{6}u^{2}+8yt^{2}u^{6}-8yu^{8}-14zt^{8}-36zt^{7}u+36zt^{6}u^{2}-36zt^{5}u^{3}+36zt^{3}u^{5}-36zt^{2}u^{6}+36ztu^{7}+14zu^{8}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.y.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 8X^{6}Y^{2}+4X^{4}Y^{4}+8X^{5}Y^{2}Z+4X^{6}Z^{2}+4X^{4}Y^{2}Z^{2}-4X^{5}Z^{3}-8X^{3}Y^{2}Z^{3}-7X^{4}Z^{4}-4X^{3}Z^{5}+8X^{2}Z^{6}+8XZ^{7}+4Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.3.y.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y^{5}z+3y^{4}z^{2}-4y^{3}z^{3}-28y^{2}z^{4}-48yz^{5}-32z^{6}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -4y^{20}z^{3}u-40y^{19}z^{4}u-108y^{18}z^{5}u-2y^{18}z^{3}u^{3}+364y^{17}z^{6}u-18y^{17}z^{4}u^{3}+3280y^{16}z^{7}u-30y^{16}z^{5}u^{3}+7932y^{15}z^{8}u+258y^{15}z^{6}u^{3}-4784y^{14}z^{9}u+1416y^{14}z^{7}u^{3}-79280y^{13}z^{10}u+1800y^{13}z^{8}u^{3}-211584y^{12}z^{11}u-7168y^{12}z^{9}u^{3}-189120y^{11}z^{12}u-34272y^{11}z^{10}u^{3}+398592y^{10}z^{13}u-54144y^{10}z^{11}u^{3}+1711872y^{9}z^{14}u+19328y^{9}z^{12}u^{3}+3264512y^{8}z^{15}u+262656y^{8}z^{13}u^{3}+4517888y^{7}z^{16}u+592896y^{7}z^{14}u^{3}+5849088y^{6}z^{17}u+761856y^{6}z^{15}u^{3}+7798784y^{5}z^{18}u+614400y^{5}z^{16}u^{3}+9175040y^{4}z^{19}u+294912y^{4}z^{17}u^{3}+7864320y^{3}z^{20}u+65536y^{3}z^{18}u^{3}+4194304y^{2}z^{21}u+1048576yz^{22}u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}y^{5}u+\frac{3}{2}y^{4}zu+\frac{1}{2}y^{3}z^{2}u+\frac{1}{4}y^{3}u^{3}-6y^{2}z^{3}u-4yz^{4}u$ |
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.
