 Class number one for discriminant minus sixty-seven

For negative discriminant $-67$, a reduced positive definite binary quadratic form satisfies $1\le a\le \sqrt{67/3}<5$ and $b$ odd with $|b|\le a$. Checking $4a\mid b^2+67$ leaves only $(a,b,c)=(1,1,17)$ under the boundary convention. The reduction algorithm for a positive definite binary quadratic form therefore puts every form in this one proper equivalence of binary quadratic forms class. Its unimodular shears give infinitely many distinct forms $x^2+(2n+1)xy+(n^2+n+17)y^2$.