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In general Sato-Tate group labels have the form $w.d.A.c.ns$, where

  • $w$ is the weight (nonnegative integer);
  • $d$ is the degree (positive integer, even if $w$ is odd);
  • $A$ is an uppercase letter that identifies the identity component among those of weight $w$ and degree $d$ (ordered by moment simplex);
  • $c$ is the number of components (positive integer);
  • $n$ is the second digit of the GAP id $[c,n]$ of the component group (positive integer);
  • $s$ is a lowercase letter used to distinguish groups for which all the preceding invariants coincide (ordered by moment simplex).

When $w=0$ we the identity component is necessarily trivial and the $A$ is omitted. When $w=0$ and $d=1$ there is exactly one Sato-Tate group for each value of $c$ and we may omit $n$ and $s$.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2021-01-18 21:57:53
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