To let $ E $ be an elliptic curve over $ mathbb {Q} $, For a prime number $ p $, To let $ mathcal {E} _p $ designate his neron model over $ mathbb {Z} _p $, Let also $ Phi_p (E) $ denote the component group of $ mathcal {E} _p $,
The structure of $ Phi_p (E) $ is known and I want to study it if $ E $ added multiplicative reduction $ p $, First, if $ E $ split multiplicative reduction at $ p $, then it is known that $ Phi_p (E) simeq mathbb {Z} / {n mathbb {Z}} $, Where $ n $ is the (normalized) $ p $adic assessment of the discriminant of $ E $, (I admit this fact.) Next, when $ E $ has an undivided multiplicative reduction $ p $, then $ Phi_p (E) simeq mathbb {Z} / {m mathbb {Z}} $, Where $ m = 1 $ or $ 2 $ so that $ m equiv n pmod 2 $,
Here's my question: Suppose that $ E $ has an undivided multiplicative reduction $ p $, As above, we use the same notation ($ m $. $ n $. $ Phi_p (E) $ Etc).
We know the following.

There is an unbranched quadratic extension $ L / { mathbb {Q} _p} $ so that $ E / { mathcal {O} _L} $ divided multiplicative reduction.

The Neron model does not change when the base changes.
If the two above are correct, then the component group $ Phi_p (E) $ can be calculated using the neron model of $ E / L $, Where $ E $ divided multiplicative reduction. Since the $ p $adic rating does not change under the unchanged extension, the component group of the Neron model of $ E / L $ is isomorphic to $ mathbb {Z} / {n mathbb {Z}} $, Consequently, $ Phi_p (E) $ is isomorphic too $ mathbb {Z} / {n mathbb {Z}} $what is wrong if $ n> 2 $,
I don't know where my argument fails. Please correct me!