三次曲线化椭圆曲线的一例

青青子衿

\[x^3+x^2 y+4 x y^2+2 y^3+2 x^2+x y+5 y^2+x+2 y+2=0\]

1. a*x^3 + b*y^3 + c*z^3 + d*x^2 y + e*x^2 z + f*y^2 x + g*y^2 z +
2.   h*z^2 x + i*z^2 y + j*x*y*z /. {x -> 2, y -> 0, z -> -1, a -> 1,
3.   b -> 2, c -> 2, d -> 1, e -> 2, f -> 4, g -> 5, h -> 1, i -> 2,
4.   j -> 1}
5. Dot[D[x^3 + x^2 y + 4 x y^2 + 2 y^3 + 2 x^2 z + x y z + 5 y^2 z +
6.      x z^2 + 2 y z^2 + 2 z^3, {{x, y, z}}] /. {x -> 2, y -> 0,
7.     z -> -1}, {x, y, z}] // Factor
8. Dot[D[x^3 + x^2 y + 4 x y^2 + 2 y^3 + 2 x^2 z + x y z + 5 y^2 z +
9.      x z^2 + 2 y z^2 + 2 z^3, {{x, y, z}}] /. {x -> 48, y -> -395,
10.     z -> 134}, {x, y, z}] // Factor
11. Inverse[{ {1, 1, 2}, {5, 4, 10}, {583846, 299818, 674653}}
12.    ].{{x}, {y}, {z}}
13. (x^3 + x^2 y + 4 x y^2 + 2 y^3 + 2 x^2 z + x y z + 5 y^2 z + x z^2 +
14.     2 y z^2 + 2 z^3) /. {x -> -((48 x)/79) - (75017 y)/493039 + (2 z)/493039,
15.     y -> 5 x - y,
16.     z -> -((134 x)/79) + (284028 y)/493039 - z/493039} /. {x ->
17.     Y/6241, y -> - X, z -> X^2} // Factor
18. {-2 X - (e + j *\[Lambda] + g* \[Lambda]^2 + d*\[Mu] +
19.       2 f*\[Lambda]*\[Mu] + 3 b*\[Lambda]^2 \[Mu])/(
20.      a + d*\[Lambda] + f*\[Lambda]^2 +
21.       b*\[Lambda]^3), \[Lambda] (-2 X - (
22.         e + j *\[Lambda] + g* \[Lambda]^2 + d*\[Mu] +
23.          2 f*\[Lambda]*\[Mu] + 3 b*\[Lambda]^2 \[Mu])/(
24.         a + d*\[Lambda] + f*\[Lambda]^2 +
25.          b*\[Lambda]^3)) + \[Mu]} /. {\[Mu] ->
26.      Y - \[Lambda]*X} /. {\[Lambda] -> (-h - 2 e X - 3 a X^2 - j Y -
27.      2 d X Y - f Y^2)/(
28.     i + j X + d X^2 + 2 g Y + 2 f X Y + 3 b Y^2)} /. {X -> -2, Y -> 0,
29.    a -> 1, b -> 2, c -> 2, d -> 1, e -> 2, f -> 4, g -> 5, h -> 1,
30.   i -> 2, j -> 1}

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\[Y^2+2497 X Y-265205 Y=X^3-1404100 X^2+6699291458 X\]

1. with(algcurves):       
2. f:=x^3+x^2*y+4*x*y^2+2*y^3+2*x^2+x*y+5*y^2+x+2*y+2:
3. v:=Weierstrassform(f,x,y,x0,y0)

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