Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves

$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}/\mathbb{Q}, E^{(k_1 k_2)}/\mathbb{Q}?$$

Here $E^{(d)}$ is the quadratic twist of $E$ by $d$. If $E$ has the short Weierstrass equation

$$\displaystyle E: y^2 = x^3 + Ax + B, A, B \in \mathbb{Z}$$

then $E^{(d)}$ has the short Weierstrass equation

$$\displaystyle E^{(d)}: dy^2 = x^3 + Ax + B.$$

In particular, let $r(E)$ denote the algebraic rank of $E$ and $r_{\text{an}}(E)$ denote the analytic rank. Are there any relations between the integers

$$\displaystyle r(E), r(E^{(k_1)}), r(E^{(k_2)}), r(E^{(k_1 k_2)})$$

or

$$\displaystyle r_{\text{an}}(E), r_{\text{an}}(E^{(k_1)}), r_{\text{an}}(E^{(k_2)}), r_{\text{an}}(E^{(k_1 k_2)})?$$

Edit:

Observe that if we write

$\displaystyle L(E,s) = \prod_p \left(1 - \frac{\alpha_p}{p^s} \right) \left(1 - \frac{\beta_p}{p^s} \right)$

for the Hasse-Weil $L$-function of $E$, then we have

$\displaystyle L(E,s) L(E^{(k_1 k_2)},s) = \prod_p \left(1 - \frac{\alpha_p}{p^s} \right) \left(1 - \frac{\chi_{k_1 k_2}(p) \alpha_p}{p^s} \right) \left(1 - \frac{\beta_p}{p^s} \right) \left(1 - \frac{\chi_{k_1 k_2}(p) \beta_p}{p^s} \right)$

and

$\displaystyle L(E^{(k_1)}, s) L(E^{(k_2)}, s) = \prod_p \left(1 - \frac{\chi_{k_1}(p) \alpha_p}{p^s} \right)\left(1 - \frac{\chi_{k_2}(p) \alpha_p}{p^s} \right)\left(1 - \frac{\chi_{k_1}(p) \beta_p}{p^s} \right)\left(1 - \frac{\chi_{k_2}(p) \beta_p}{p^s} \right).$

In particular, their Euler factor at $p$ coincides whenever $\chi_{k_1 k_2}(p) = -1$, and whenever $\chi_{k_1 k_2}(p) = \chi_{k_1}(p) = \chi_{k_2}(p) = 1$. Therefore, the Euler factors match for $3/4$'s of primes $p$.