I want to construct an explicit elliptic curve $E$ over a number field $K$ such that $Gal(K(E[l])/K) \cong D_{10}$ where $D_{10}$ is the dihedral group of order 10 and $l$ is a prime number.
Preferably, I would like the degree $[K:\mathbb{Q}]$ to be small and $l=5$ as I wish to run some computations on it.
So far, I know $K$ cannot be $\mathbb{Q}$ and was but was hoping to find a mod 5 representation of a CM curve over $\mathbb{Q}$ with image isomorphic to $G_{20} = \langle a,b | a^5, b^4, bab^{-1}=a^2 \rangle$ and then base change the curve to the real subfield of $\mathbb{Q}\left( e^{2\pi i/5} \right)$ but can't find such a curve to get a $D_{10}$ image.
Can anyone help?
Take $E: y^2+y=x^3-x^2$, aka 11a3 in Cremona's tables. Then, the image of the representation mod $5$ is precisely given by all the matrices of the form $$\left(\begin{array}{cc} 1 & b\\ 0 & c \end{array}\right)$$ where $b\in \mathbb{Z}/5\mathbb{Z}$ and $c\in (\mathbb{Z}/5\mathbb{Z})^\times$. I know this, because the type of image is listed here. Thus, the image is a group $G_5$ of order $20$, isomorphic to the semi-direct product of $\mathbb{Z}/5\mathbb{Z}\rtimes \mathbb{Z}/4\mathbb{Z}$. In particular, $G_5$ has a subgroup isomorphic to $D_{10}$ which fixes a (the) quadratic field $K$ contained in $\mathbb{Q}(E[5])$. Since $\mu_5$ needs to be contained in $\mathbb{Q}(E[5])$, the fixed field of $D_{10}$ must be $K=\mathbb{Q}(\sqrt{5})$.
Hence, with this construction, if you base extend $E/K$, then $\operatorname{Gal}(K(E[5])/K)\cong D_{10}$.