You have a system of 52 polynomial equations in 28 unknowns, of high degree. The first 26 equations, however, are very easy. You can solve them and replace in the remaining equations:
sol1 = Solve[0 == Union@Eq[[1 ;; 26]]]
remainingFirstStep = Complement[Eq /. First@sol1, {0}]
The first two remaining equations are linear, and you can solve for them and replace in the rest:
sol2 = Solve[remainingFirstStep[[1 ;; 2]] == 0]
remainingSecondStep =
Factor[remainingFirstStep[[3 ;; -1]] /. First@sol2]
The fifteenth and the next-to-last of the remaining equations factor into linear polynomials. Solve these equations and replace into the rest:
sol3 = Solve[remainingSecondStep[[{15, -2}]] == 0]
remainingThirdStep =
Factor@Map[Complement[#, {0}] &, Factor[remainingSecondStep /. sol3]]
You get two sets of equations. These are easy enough to solve directly:
sol4 = Map[Solve[# == 0] &, remainingThirdStep]
It seems that the only nonzero solution is sol4[[2,1]]:
{a[13] -> -(a[18]/2), a[19] -> a[18]}
You can check this solution by replacing into the original equations:
nonzeroSol =
Solve[Flatten[{sol4[[2, 1]], sol3[[2]], sol2, sol1} /. Rule -> Equal]]
Expand[Eq /. nonzeroSol]
