Class numbers of quadratic Diophantine equations

•Define the class number of quadratic Diophantine equations over algebraic number fields.•Prove a mass formula for quadratic Diophantine equations.•Give a method for computing the class number of quadratic Diophantine equations by using the mass formula.

In this paper, the class number of a quadratic Diophantine equation is defined so that it can be viewed as a measure of the obstruction of the local-global principal for quadratic Diophantine equations, and we also give a method for computing the class number of quadratic Diophantine equations by using the mass formula.