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The matrix you constructed by hand doesn't look like a Hessian to me. Here is the Hessian:

sage: H = L.function(x,y,l).hessian()(x,y,l)
sage: H

$$\left(\begin{array}{rrr} A {\left(\alpha - 1\right)} \alpha x^{\alpha - 2} y^{\beta} & A \alpha \beta x^{\alpha - 1} y^{\beta - 1} & -p_{x} \\ A \alpha \beta x^{\alpha - 1} y^{\beta - 1} & A {\left(\beta - 1\right)} \beta x^{\alpha} y^{\beta - 2} & -p_{y} \\ -p_{x} & -p_{y} & 0 \end{array}\right)$$