Paris Brown, 17, had her mobile phone seized and was interviewed under caution by Special Branch investigating the material, they said.
The teenager was forced to stand down from her £15,000-a-year-role as Kent Police's youth police and crime commissioner earlier this month following publication of the tweets.
Today it emerged that Paris's lawyers, Olswang, have written to Kent Police Chief Constable Ian Learmonth about the scope and nature of the investigation, including the decision to seize her phone and for Special Branch to quiz her.
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Samson the miniature steam train has re-entered service, on the Romney Hythe and Dymchurch railway, Kent's mainline in miniature.
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The ABP Southampton Marathon is the city's biggest running event, attracting 10,000 runners - and walkers - and thousands of spectators.