Math of Logistic regression cost function

In the current scikit-learn documentation for binary Logistic regression there is the minimization of the following cost function:

$$\min_{w, c} \frac{1}{2}w^T w + C \sum_{i=1}^n \log(\exp(- y_i (X_i^T w + c)) + 1)$$

Questions:

• what is the $$c$$ term? It is not explained in the documentation
• What is the cost function minimized when LogisticRegression(multiclass=multinomial) is used instead?