Homogenous space of elliptic curve E/$Bbb Q$

No, this is not true. Consider the curve $$C : 2w^2 = 4 - 20z^4$$ which has Jacobian $$E : y^2 = x^3 + 5x$$ with an isomorphism $E to C$ over $mathbb{Q}(sqrt{5})$ given by $$ (x, y) to left(frac{sqrt{2}x}{y}, sqrt{2}left(x - frac{5}{x}left(frac{x}{y}right)^2 right) right)$$ for details see Silverman X3.7.

I claim that $C$ has no $5$-adic points. To see this note that it suffices to show that there are no $u, v, w in mathbb{Z}_5$ such that $$w^2 = 2u^4 - 10v^4$$

Suppose otherwise, then we may assume that $(u, v, w) = (1)$. Then $u,w notequiv 0 pmod{5}$, hence $$w^2 equiv 2u^4 pmod{5}$$ but in that case $2 equiv (w/u^2)^2 pmod{5}$, a contradiction since $2$ is not a square in $mathbb{F}_5$.

Thus $C(mathbb{Q}_p) = emptyset$.

The standard reference for such things is Silverman Chapter X.