Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x u + z t + w t + w u $ |
| $=$ | $2 x t + x u - z u$ |
| $=$ | $2 x^{2} - x z + x w + z^{2} + z w$ |
| $=$ | $x^{2} + x z - x w + 2 y^{2} - 2 z^{2} + z w + t^{2} + t u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{8} - 28 x^{6} z^{2} + 16 x^{4} y^{4} - 27 x^{4} y^{2} z^{2} + 105 x^{4} z^{4} + 144 x^{2} y^{4} z^{2} + \cdots + 196 z^{8} $ |
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Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ -135 x^{3} z - 78 x^{2} y z + 39 x z^{3} + 10 y z^{3} $ |
| $0$ | $=$ | $-3 x^{2} + y^{2} + z^{2}$ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3^4}\cdot\frac{26873856zw^{9}u^{2}+6718464zw^{7}u^{4}+53187840zw^{5}u^{6}+59486400zw^{3}u^{8}+137308608zwu^{10}+2985984w^{12}+8957952w^{10}u^{2}-8957952w^{8}u^{4}+13810176w^{6}u^{6}-2297808w^{4}u^{8}+57660984w^{2}u^{10}+527345t^{12}+6401868t^{11}u+31868028t^{10}u^{2}+85008032t^{9}u^{3}+129514179t^{8}u^{4}+113894232t^{7}u^{5}+71836258t^{6}u^{6}+29530236t^{5}u^{7}+1911738t^{4}u^{8}+21120472t^{3}u^{9}+13233168t^{2}u^{10}+15248424tu^{11}+3437117u^{12}}{u^{4}(497664zw^{5}u^{2}+404352zw^{3}u^{4}-97200zwu^{6}+82944w^{8}+228096w^{6}u^{2}-55728w^{4}u^{4}-130248w^{2}u^{6}-1024t^{8}+4096t^{7}u-1280t^{6}u^{2}+7936t^{5}u^{3}+43793t^{4}u^{4}+9764t^{3}u^{5}+14656t^{2}u^{6}-3592tu^{7}+2195u^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.3.cm.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{8}+16X^{4}Y^{4}-28X^{6}Z^{2}-27X^{4}Y^{2}Z^{2}+144X^{2}Y^{4}Z^{2}-108Y^{6}Z^{2}+105X^{4}Z^{4}+816X^{2}Y^{2}Z^{4}+1044Y^{4}Z^{4}-196X^{2}Z^{6}-1347Y^{2}Z^{6}+196Z^{8} $ |
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.
