Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} + x y - y^{2} - w^{2} - t^{2} $ |
| $=$ | $x w - x t + y w - z t$ |
| $=$ | $ - 2 x t + y t + z w$ |
| $=$ | $x^{2} - x z - z^{2} + w^{2} + t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} + 9 x^{4} z^{2} - 3 x^{3} y^{2} z - 6 x^{3} z^{3} - 6 x^{2} y^{2} z^{2} + 10 x^{2} z^{4} + \cdots + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} + 3x^{5} + 5x^{4} + 5x^{2} - 3x - 1 $ |
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This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{5^2}\cdot\frac{328125xz^{7}-12437500xz^{5}t^{2}+319500000xz^{3}t^{4}-6691342500xzt^{6}+3328125y^{2}z^{4}t^{2}-95602500y^{2}z^{2}t^{4}+2074561875y^{2}t^{6}+1781250yz^{5}t^{2}-40651875yz^{3}t^{4}+731857500yzt^{6}+203125z^{8}-9390625z^{6}t^{2}+265663125z^{4}t^{4}-5856576875z^{2}t^{6}+9261w^{8}+142884w^{7}t+1621242w^{6}t^{2}+14187312w^{5}t^{3}+109246545w^{4}t^{4}+759812688w^{3}t^{5}+8144808742w^{2}t^{6}+745762116wt^{7}+8037172386t^{8}}{(w^{2}+t^{2})^{4}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.48.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y^{2}-3X^{3}Y^{2}Z+9X^{4}Z^{2}-6X^{2}Y^{2}Z^{2}-6X^{3}Z^{3}+3XY^{2}Z^{3}+10X^{2}Z^{4}+Y^{2}Z^{4}-6XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.48.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w^{2}t-\frac{1}{3}wt^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{3}zw^{6}t^{2}+\frac{11}{9}zw^{5}t^{3}+\frac{35}{27}zw^{4}t^{4}-\frac{20}{9}zw^{3}t^{5}+\frac{5}{9}zw^{2}t^{6}+\frac{1}{9}zwt^{7}-\frac{1}{27}zt^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle wt^{2}-\frac{1}{3}t^{3}$ |
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.
