Covid-19 test results that will begin to roll in on Tuesday will be crucial in ascertaining whether the virus has spread, health authorities say.
Among the results will be those from about 30 Papatoetoe High School students and staff close contacts of a confirmed Covid-19 case in the community.
The teenage girl and her parents were announced as positive cases on Sunday.
That news led to Auckland being put into alert level 3, and the rest of the country into alert level 2.
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The trio are now in the Auckland quarantine facility, Jet Park.
Thousands lined up for Covid-19 tests at a pop-up facility at Papatoetoe High School on Monday.
Thousands of members of the Papatoetoe High School community were tested on Monday, with some people waiting up to four hours to be swabbed.
Those at the back of the line were told to come back on Tuesday after it became clear health workers could not fit them all in.
Director-General of Health Dr Ashley Bloomfield said it was encouraging to see the extent of testing being carried out, but more results are needed to really rule out onward transmission, or further transmission.
Results over the coming days would give authorities an increasing level of assurance, assuming they are all negative, he said.
Meanwhile, investigations into the source of infection for the three new cases, and whether there has been any further spread, is ongoing.
On Monday, Prime Minister Jacinda Ardern confirmed the family have the UK variant of the virus.
Work is underway to find the source of the case after testing has not linked it to any cases in managed isolation and quarantine facilities (MIQ).
Serology testing is also being carried out at Papatoetoe High School and among the mothers co-workers at LSG Sky Chefs.
Prime Minister Jacinda Ardern says investigations will continue on Tuesday.
Ardern said authorities continue to pursue a range of theories, even some considered unlikely or highly unlikely as it is important to leave no stone unturned.
An alert level change will not be discussed on Tuesday.
On Monday, Ardern said she could say, hand on heart, officials had not yet had any conversations on the possibility of an extension after Wednesday.
[You can get a] little sense early on, but [there are] no conclusions yet.