It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2in [0,9]$ and the eccent
ricity $ein [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(bb,e)$ range $[0,9]times [0,1)$ via the $om$-index theory of symplectic paths for $om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $om$-index decreasing property of the solutions in $bb$ for fixed $ein [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $om$-degeneracy curves for $om ot=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.
