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The database currently contains all rational Sato-Tate groups of weight 1 and degree up to 6, as well as all Sato-Tate groups of weight 0 and degree 1 with component group of order at most $10^{20}$. Here are some further statistics.

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By weight: 0  1 
By degree: 1  2  3  4  5  6 
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e.g. 0.1.3 or 0.1.mu(3), or 1.2.B.2.1a or N(U(1)), or 1.4.A.1.1a or 1.4.USp(4)