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In this paper, we give an explicit formula as well as a practical algorithmfor computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times\text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve underthe assumption that all points in $J[2]$ are $K$-rational. We also give anexplicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow\text{Br}(K)$ under the same assumption. Finally, we include a worked exampledemonstrating we can indeed improve the rank bound given by a 2-descent viacomputing the Cassels-Tate pairing.

Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points