The EDS manages one cookie. This cookie is called
_saml_idp and its use and format is described in section 4.3.1 Common Domain Cookie of the SAML2 profile specification. The function of the cookie is to record previously selected IdPs. The cookie is persistent with a lifetime controlled by the
samlIdPCookieTTL setting as described here.
Additionally in versions from 1.2 you can specify a second cookie name to control 'Auto Follow'