Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda\_{E}^{alg} $, $ \mu\_{E}^{alg} $, $ \lambda\_{E}^{an} $ and $ \mu\_{E}^{an} $ at $p$. 


The "algebraic" Iwasawa invariants $ \lambda\_{E}^{alg} $ and $ \mu\_{E}^{alg} $ are defined in terms of the structure of the $p$-primary subgroup $ Sel\_{E}(\mathbb{Q}\_{\infty})\_{p} $ of the Selmer group for $E$ over the cyclotomic $ \mathbb{Z}\_{p} $-extension $ \mathbb{Q}\_{\infty} $ of $\mathbb{Q}$. The definition of the "analytic" invariants $ \lambda\_{E}^{an} $ and $ \mu\_{E}^{an} $ is in terms of the $p$-adic $L$-function for $E$ constructed by Mazur and Swinnerton-Dyer.


Now the Main Conjecture (Mazur) implies that $ \mu\_{E}^{alg}=\mu\_{E}^{an} $ and $ \lambda\_{E}^{alg}=\lambda\_{E}^{an} $.


Let $E\_1$ and $E\_2$ are two elliptic curves over $\mathbb{Q}$ such that there exists a $\mathbb{Q}$-isogeny from $E\_1$ to $E\_2$. Is the Main Conjecture invariant under a $\mathbb{Q}$-isogeny i.e, if the conjecture is true for any one of the curves then it is also true for the other one $?$



Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny
Yes, the main conjecture is isogeny-invariant. See here:


B. Perrin-Riou, *Variation de la fonction $L$ $p$-adique par isogénie*, Algebraic number theory, Adv. Stud. Pure Math. **17** (1989), pp. 347-358.


