Joint region and coefficient covariance

Consider the simple linear regression model $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, $i = 1, 2,…, n$, where $\beta_0, \beta_1$ are unknown parameters, $\epsilon_i$'s are uncorrelated random errors with mean 0 and finite variance $\sigma^2 > 0$. Let $\hat\beta_i$ be the least squares estimator of $\beta_i$, $i = 0,1$. Consider the following statements: (I) A 95% joint confidence region for $(\beta_0, \beta_1)$ is the region bounded by an ellipse. (II) The expression for covariance between $\hat\beta_0$ and $\hat\beta_1$ does not involve $\sigma^2$. Which of the above statements is/are true?