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Additionally, you need to have alerts assigned with Manage access (see Assign resources to a role), which will be those you will see on the list.
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About triggered alert priority
The Priority column indicates the priority level assigned to the alert definition when created. However, circumstances can change when triggered and may consequently be considered different in priority. There are five possible values, each being assigned a numerical value that will be displayed in auditing tables:
Very low (0)
Low (3)
Normal (5)
High (7)
Very high (10)
Changing priority from the alert list
Changing priority of a single alert
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