flow
#flow
#flow
\Delta(a,b,c,d,e) = 256a^3e^3 - 192a^2bde^2 - 128a^2c^2e^2 + 144a^2cd^2e - 27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e - 80abc^2de
+ 18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 + 18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2d^2
#flow,left
#flow,left
\Delta(a,b,c,d,e) = 256a^3e^3 - 192a^2bde^2 - 128a^2c^2e^2 + 144a^2cd^2e - 27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e - 80abc^2de
+ 18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 + 18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2d^2
#flow,indent,rowgap=1em
#flow,indent,rowgap=1em
\Delta(a,b,c,d,e) = 256a^3e^3 - 192a^2bde^2 - 128a^2c^2e^2 + 144a^2cd^2e - 27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e - 80abc^2de
+ 18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 + 18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2d^2
#flow,left,indent,fold=500
#flow,left,indent,fold=500
\Delta(a,b,c,d,e) = 256a^3e^3 - 192a^2bde^2 - 128a^2c^2e^2 + 144a^2cd^2e - 27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e - 80abc^2de
+ 18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 + 18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2d^2
#list
#list
\alpha, \frac 1 {\alpha}, 1 - \alpha, \frac {\alpha} {\alpha - 1}, \frac {\alpha - 1} {\alpha}, \frac 1 {1 - \alpha},
\beta, \frac 1 {\beta}, 1 - \beta, \frac {\beta} {\beta - 1}, \frac {\beta - 1} {\beta}, \frac 1 {1 - \beta},
\lambda, \frac 1 {\lambda}, 1 - \lambda, \frac {\lambda} {\lambda - 1}, \frac {\lambda - 1} {\lambda}, \frac 1 {1 - \lambda}
fold
#fold
#fold
\sqrt[30] {\discrim(a,b,c,d,e,f,g)}\bigint_{C} \frac 1 {\sqrt[3]{ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g}} dx =
\sqrt[30] {\discrim(p,q,r,s,t,u,v)} \bigint_{C'} \frac 1 {\sqrt[3]{py^6 + qy^5 + ry^4 + sy^3 + ty^2 + uy + v}} dy
#fold,end
#fold
\sqrt[30] {\discrim(a,b,c,d,e,f,g)} \bigint_{C} \frac 1 {\sqrt[3]{ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g}} dx = $
\sqrt[30] {\discrim(p,q,r,s,t,u,v)} $ \bigint_{C'} \frac 1 {\sqrt[3]{py^6 + qy^5 + ry^4 + sy^3 + ty^2 + uy + v}} dy
#fold
#fold
\sqrt[30] {\discrim(a,b,c,d,e,f,g)} $ \bigint_{C} \frac 1 {\sqrt[3]{ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g}} dx = $
\sqrt[30] {\discrim(p,q,r,s,t,u,v)} $ \bigint_{C'} \frac 1 {\sqrt[3]{py^6 + qy^5 + ry^4 + sy^3 + ty^2 + uy + v}} dy
#fold2 with 2 parts - doesn't look good
#fold2
\sqrt[30] {\discrim(a,b,c,d,e,f,g)} $ \bigint_{C} \frac 1 {\sqrt[3]{ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g}} dx = $$
\sqrt[30] {\discrim(p,q,r,s,t,u,v)} $ \bigint_{C'} \frac 1 {\sqrt[3]{py^6 + qy^5 + ry^4 + sy^3 + ty^2 + uy + v}} dy
#fold2 with 3 parts - doesn't look good
#fold2,debug
\beta \space = \space \zeta^i \alpha, \quad$ \frac 1 {\zeta^i \alpha}, \quad$ \zeta^j \frac {1 - \zeta^i\alpha} {1 + \zeta^i\alpha} $$
\qquad \textsf{where} \qquad $$
i=0\ldots 3, \enspace j=0 \ldots 3
train
#train,width=1200,flow,fold
#train,width=1200,flow,fold
\small{\begin{vmatrix} a & b & c \\ p & q & r \\ q & r & s \\ \end{vmatrix}} $
= $ a(qs - r^2) + b(qr - ps) + $ c(pr - q^2) $$
= $ - \tfrac 1 {18} ap^2\big[ $
(\alpha - \mu)(\beta - \mu)(\nu - \omega)^2 + $
(\alpha - \nu)(\beta - \nu)(\mu - \omega)^2 + $
(\alpha - \omega)(\beta - \omega)(\mu - \nu)^2\big]
#train,width=1200,flow,fold=430
#train,width=1200,flow,fold=430
\small{\begin{vmatrix} a & b & c \\ p & q & r \\ q & r & s \\ \end{vmatrix}} $
= $ a(qs - r^2) + b(qr - ps) + c(pr - q^2) $$
= $ - \tfrac 1 {18} ap^2\big[ $
(\alpha - \mu)(\beta - \mu)(\nu - \omega)^2 + $
(\alpha - \nu)(\beta - \nu)(\mu - \omega)^2 + $
(\alpha - \omega)(\beta - \omega)(\mu - \nu)^2\big]