What is the lower $n$ such that $\mathbb{CP}^6$ can be embedded in $\mathbb{R}^n$?

Question

Using certain integrality theorem I am obtaining that $\mathbb{CP}^6$ cannot be embedded in a Euclidean space of dimension 19 or less.

Also it is well known that $\mathbb{CP}^6$ can be immersed into $\mathbb{R}^{23}$.

Then my question is:

What is the lower $n$ such that $\mathbb{CP}^6$ can be embedded in $\mathbb{R}^n$?