Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + 2 x y + 2 x z + x w - y^{2} - 2 y z - z^{2} + w^{2} $ |
| $=$ | $x^{2} + x y + x z - x w + y^{2} + 2 y z - y w + z^{2} - z w - 2 w^{2} + 2 t^{2}$ |
| $=$ | $3 x^{2} - 2 x w + 2 y^{2} - 3 y z + y w + 2 z^{2} + z w - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{7} - 7 x^{6} z + 21 x^{5} z^{2} - 21 x^{4} z^{3} - 98 x^{3} y^{2} z^{2} + 35 x^{3} z^{4} + \cdots - 58 z^{7} $ |
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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 112 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^7}\cdot\frac{(7w^{2}-6t^{2})^{3}(6117748xw^{7}-5243784xw^{5}t^{2}+1137584xw^{3}t^{4}-50400xwt^{6}-8115380yzw^{6}+5814536yzw^{4}t^{2}-924336yzw^{2}t^{4}+18144yzt^{6}+11764900yw^{7}-11126920yw^{5}t^{2}+2815344yw^{3}t^{4}-163296ywt^{6}+11764900zw^{7}-11126920zw^{5}t^{2}+2815344zw^{3}t^{4}-163296zwt^{6}+11762499w^{8}-20299426w^{6}t^{2}+9779812w^{4}t^{4}-1374296w^{2}t^{6}+26624t^{8})}{t^{14}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.112.5.bf.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x-2y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z+w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{7}-7X^{6}Z+21X^{5}Z^{2}-98X^{3}Y^{2}Z^{2}-21X^{4}Z^{3}+98X^{2}Y^{2}Z^{3}-196Y^{4}Z^{3}+35X^{3}Z^{4}+98XY^{2}Z^{4}-35X^{2}Z^{5}+196Y^{2}Z^{5}-49XZ^{6}-58Z^{7} $ |
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.
