Officers have appealed for witnesses and asked for information on a Falcon Storm mountain bike with straight handlebars and distinctive yellow/green front fork.
Superintendent Eamonn Curley told reporters on Thursday: "Where the crime occurred, is popular among Tullamore residents and is widely used for recreational purposes.
"At this stage, there is no information that the victim and her killer were known to each other. We believe this crime was committed by one male who acted alone."
He added: "Our prayers are with Ashling's family at this time and indeed the wider community, her teaching colleagues, and the children she taught who are without their teacher today."
Murphy taught 28 pupils at Durrow National School and was a member of the National Folk Orchestra of Ireland.
Her principal James Hogan said she was a "fantastic, beautiful, an extraordinary sportsperson, Irish musician, choral conductor and an inspiration".
Declan Harvey, Fianna Fail councillor for the area, said: "I am devastated. I couldn't sleep last night thinking of it all because it is a place I go all the time."
The route by the canal is called Fiona's Way, named in memory of a missing woman Fiona Pender.
"She went there jogging, she does it regularly. I am lost for words. Tullamore is probably the safest place in the world, and after yesterday people will be nervous to go down to the canal," Harvey said.
Justice Minister Helen McEntee said: "My thoughts tonight are with the family of the young woman killed in a truly shocking crime, with her friends and with her community."
Minister of State Josepha Madigan for Special Education and Inclusion said: "A young woman can't even go for a run along a canal pathway - at four o'clock, in the middle of the day - only to be attacked and murdered.
"We need to urgently address why we live in a country where this can happen."
Giving an update into the investigation, gardai said: "The male who was arrested and detained by An Garda Siochana investigating this fatal assault has now been released from this detention.
"This male has been eliminated from garda inquiries and is no longer a suspect.
"An Garda Siochana continue to appeal for any information on a Falcon Storm mountain bike with straight handlebars and distinctive yellow/green front forks.
"Gardai continue to appeal for any person with any information on this fatal assault, who may not already have spoken to investigators, to come forward with that information.
"Gardai continue to appeal to any persons that were in the Cappincur/Canal walk area of Tullamore, County Offaly before 4pm, yesterday afternoon Wednesday, January 12, 2022, to make contact with them.
"Anyone with information is asked to contact Tullamore Garda Station on 057 932 7600, the Garda Confidential Line on 1800 666 111, or any garda station."
