Does it matter whether we put regularization parameter ($C$) with error or weight term in Kernel ridge regression?

Question

Kernel ridge regression associate a regularization parameter $C$ with weight term ($\beta$):

$\text{Minimize}: {KRR}=C\frac{1}{2} \left \|\beta\right\|^{2} + \frac{1}{2}\sum_{i=1}^{\mathcal{N}}\left\|e_i \right \|_2^{2} \\ \text{Subject to}:\ {\beta^T\phi_i}=y_i - e_i, \text{ }i=1,2,...,\mathcal{N}$

If we associate $C$ with an error term as follows:

$\text{Minimize}: {KRR}=\frac{1}{2} \left \|\beta\right\|^{2} + C\frac{1}{2}\sum_{i=1}^{\mathcal{N}}\left\|e_i \right \|_2^{2} \\ \text{Subject to}:\ {\beta^T\phi_i}=y_i - e_i, \text{ }i=1,2,...,\mathcal{N}$

then how this second formulation is different from the first one?

or

Can we associate $C$ either with weight term or error term in Kernel ridge regression?