$V^7$ is the set of length $7$ codewords. $V^7[x] = \mathbb F_2[x]/ \langle x^7-1\rangle $.
I know that each of these cyclic codes is constructed from a canonical generator (divisor of $x^7-1$).
So here's my set of divisors:
$$D=\{x^7-1,\; 1,\; 1+x,\; 1+x+x^3,\; 1 +x^2+x^3,\; (1+x)(1+x+x^3),\; (1+x)(1+x^2+x^3),\; (1+x+x^3)(1+x^2+x^3)\}$$
Each of these generates a different cyclic code of length $7$.
And I know that $\langle x^7-1 \rangle$ will correspond to the word $0000000$, since we always mod by $x^7-1$ after multiplying the elements of $V^7[x]$ by $x^7-1$.
Also $\langle 1 \rangle$ will just give $V^7$ since we multiply by $1$ then mod $x^7-1$ for each element in $V^7[x]$.
But what about the other ones? I don't see an obvious way of describing them without manually computing the ideals (which definitely is not the efficient way of solving this).