Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto an elliptic curve. It's easy to see that $C$ is a genus $8$ curve that is not hyperelliptic, so the canonical linear system gives an embedding into $\mathbb{P}^8$. I want to explicitly construct and understand this situation. My approach is to consider the graded ring
$\bigoplus_{n\geq0}H^0(C,nD+np)$
I start with the basis $\{ux_0,ux_1,ux_2\}$ of $H^0(C,D+p)$. Here $u:\mathcal{O}_C\hookrightarrow\mathcal{O}_C(p)$ is the constant section. Passing to degree 2 one should get 8 generators and the second symmetric power of the basis just given has dimension 6, so we need two extra generators to complete a basis for $H^0(C,K_C)$. I have several guesses about how these should look like in terms of the $x_i$'s and with a bit more of work one sees that passing to degree $3$ would require another $2$ new generators in degree $3$. I degree 3 we can see already the equation of the elliptic curve $E$ that we may assume to be
$ac^2=b^3-\alpha ab^2+\beta a^2b$;
Where $a:=ux_0,b:=ux_1,c:=ux_2$ and some constants $\alpha,\beta$. How can I see the double cover branched at 14 points in terms of $a,b,c$ and my new generators in degrees 2 and 3? Name them $y_0,y_1\in H^0(C,K_C)$ and $z_0,z_1\in H^0(C,3D+3p)$. Any useful observation will be greatly appreciated!
