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We construct the general algebraic curve of degree four solving the classicalsigma model on RxS5. Up to two loops it coincides with the algebraic curve forthe dual sector of scalar operators in N=4 SYM, also constructed here. Weexplicitly reproduce some particular solutions.Comment: 58 pages, v2: minor changes, references added, v3: "length"  identified in sigma model, comparison of curves improved, minor changes, v4:  references added, to appear in Commun. Math. Phy

Algebraic Curve for the SO(6) Sector of AdS/CFT