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本帖最后由 青青子衿 于 2024-4-29 02:45 编辑
\begin{gather*}
\begin{split}
y^2&=x^4+a x^3+b x^2+c x+d\\
v^2&=u^3+A u+B\\
q^2&=p^3+A p+B\\
\end{split}\\
\\
\left\{\begin{split}
x&=\frac{v+q}{u-p}-\frac{a}{4}\\
y&=2u+p-\left(\frac{v+q}{u-p}\right)^2\\
b&=\frac{3}{8} \left(a^2-16 p\right)\\
c&=\frac{1}{16} \left(a^3-48 a p-128 q\right)\\
d&=\frac{1}{256} \left(a^4-96 a^2 p-512 a q-1024 A-768 p^2\right)\\
\end{split}\right.\\
\\
\left\{\begin{split}
u&=\frac{1}{12} \left(3 a x+b+6 x^2+6 y\right)\\
v&=\frac{1}{8} \left(3 a x^2+a y+2 b x+c+4 x^3+4 x y\right)\\
A&=-\frac{1}{48} \left(b^2-3 a c+12 d\right)\\
B&=\frac{1}{1728}(27 a^2 d-9 a b c+2 b^3-72 b d+27 c^2)\\
p&=\frac{1}{48} \left(3 a^2-8 b\right)\\
q&=-\frac{1}{64} \left(a^3-4 a b+8 c\right)\\
\end{split}\right.
\end{gather*}
Multiples of Points on Elliptic Curves and Continued Fractions
researchgate.net/publication/31389206
- d + c x + b x^2 + a x^3 + x^4 -
- y^2 /. {x -> (v - 1/64 (a^3 - 4 a*b + 8 c))/(
- u - 1/48 (3 a^2 - 8 b)) - a/4,
- y -> 2 u + (3 a^2 - 8 b)/
- 48 - ((v - 1/64 (a^3 - 4 a*b + 8 c))/(
- u - 1/48 (3 a^2 - 8 b)))^2} /. {a -> 12, b -> 942, c -> -17604,
- d -> -2151} // Factor
- u^3 + A*u + B - v^2 /. {u -> (b + 3 a*x + 6 x^2 + 6 y)/12 ,
- v -> (c + 2 b*x + 3 a*x^2 + 4 x^3 + a*y + 4 x*y)/8 ,
- A -> -((b^2 - 3 a c + 12 d)/48) ,
- B -> (2 b^3 - 9 a b c + 27 c^2 + 27 a^2 d - 72 b d)/1728} // Factor
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