Example of $E_0(K)$ which satisfies $[E(K):E_0(K)]＝2$

Question

Let $E$ be an elliptic curve over local field $K$ which has good reduction. $E(K)/E_0(K)$ is known to be finite .

Now I'm looking for an example of $E_0(K)$ which satisfies $[E(K):E_0(K)]＝2$.

Let $K＝\Bbb Q_p$, I tried some elliptic curve like $E:y^2＝x^3＋3$ over $\Bbb Q_3$, $E(\Bbb Q_3)＝4$,but I couldn't list the points of $E_0(\Bbb Q_3)$.

Thank you in advance.

Answer 1

It's pretty easy to construct such examples. By a Theorem of Kodaira-Neron (Silverman, Theorem VII.6.1) if $K$ is a finite extension of $\mathbb{Q}_p$ and $E$ has split multiplicative reduction then the Tamagawa number $[E(K) : E_0(K)]$ is equal to the valuation of the discriminant (in specific cases you can just follow Tate's algorithm to verify).

Just looking through the LMFDB one can find examples pretty quickly. For example the curve with Cremona Label $21a2$ $$E : y^2 + xy = x^3 - 49x - 136.$$

has split multiplicative reduction at $3$ and discriminant $3^2 \cdot 7^4$.