Selmer ranks under quadratic twists satisfying the Heegner hypothesis

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit formulae describing the effect of twisting on Selmer ranks in terms of matrices over $\mathbb{F}_{2}$. As an application, we show that these formulae are compatible with predictions made by the parity conjecture.