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We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \\prod_{i = 1}^{2g + 1} (x - \\alpha_{i})$. Our main result states that the image of Galois in $\\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $\\Gamma(2) \\lhd \\mathrm{Sp}(T_{2}(J))$. 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We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \\prod_{i = 1}^{2g + 1} (x - \\alpha_{i})$. Our main result states that the image of Galois in $\\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $\\Gamma(2) \\lhd \\mathrm{Sp}(T_{2}(J))$. 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