The government and opposition are playing down the possibility of a referendum to amend section 44 of the constitution and resolve the citizenship crisis, instead urging political candidates to be more diligent in ensuring they are eligible for Parliament.

The months-long saga ended the careers of another five MPs on Wednesday, with the High Court ruling Labor senator Katy Gallagher ineligible and triggering the rapid resignations of four lower house MPs. A 'super Saturday' of five byelections – including one stemming from another Labor MP's unrelated resignation – now looms as a test for both Prime Minister Malcolm Turnbull and Opposition Leader Bill Shorten.

Labor MPs Justine Keay, Josh Wilson and Susan Lamb and Centre Alliance MP Rebekha Sharkie announced their resignations on Wednesday afternoon soon after the court ruled that former senator Katy Gallagher was still a dual citizen when she was formally nominated at the last election.

Susan Lamb, Josh Wilson, Justine Keay, Rebekha Sharkie and Tim Hammond (who resigned last week for family reasons). Credit:Alex Ellinghausen

Asked how the Parliament would solve the crisis that began in mid-2017, Mr Turnbull told ABC radio the government had previously argued for a more lenient interpretation of section 44 but it was "very hard" to change the constitution.