  In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of
number fields $L/K$ ($p\geq 5$) for the twist $1\oplus \eta \oplus \tau $:
W(E/K,1\oplus \eta \oplus \tau)=(-1)^{< 1\oplus\eta \oplus \tau, X_{p}(E/L)>},
where $E$ is an elliptic curve over $K,$ $\eta$ and $\tau$ are respectively the
quadratic character and an irreductible representation of degree 2 of
$Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main
novelty is that we use a congruence result between $% \epsilon_{0}$-factors
(due to Deligne) for the determination of local root numbers in bad cases
(places of additive reduction above 2 and 3). We also give applications to the
$p$-parity conjecture (using the machinery of the Dokchitser brothers).
