=(2+2k^2/( 1+2k^2))^2+(k/( 1+2k^2)+(5k^2+2)/(k( 1+2k^2)))^2

=((2+6k^2)/( 1+2k^2))^2+((6k^2+2)/(k( 1+2k^2)))^2

=(2+6k^2)^2 *(k^2+ 1)/(k( 1+2k^2))^2)

So PC = 2 (1+3k 2) root number (k 2+ 1)/| k | root number (1+2k 2).

The second to third steps are to multiply the numerator denominator of the first formula by k 2 at the same time.