Parity of ranks of elliptic curves with equivalent mod 𝑝 Galois representations

Given two elliptic curves E 1 E_1 and E 2 E_2 defined over the field of rational numbers Q \mathbb {Q} that have good and ordinary reduction at an odd prime p p , and have equivalent, irreducible mod p p Galois representations, we study the variation of the parity of Selmer ranks and analytic ranks of E 1 E_1 and E 2 E_2 over certain number fields.