England team celebrates a wicket.
In the fifth over of the day, Jonny Bairstow (101) went on to notch his fifth Test century while Jack Leach (16) played a handy knock as England crossed the 300-run mark.
New Zealand's position became more hard when Stuart Broad and James Anderson, bowling their best new ball spells in some time, generated swing and reduced the home side to 17-4, then 36-5 with all of its top batsmen out.
England's top two wicket-takers in Tests set a milestone of opening the bowling in the format for the 150th time.
New Zealand slumped to 36-5 in its first innings with all of its recognized batsmen dismissed before wicketkeeper B.J. Watling and allrounder Colin de Grandhomme combined in a record partnership for New Zealand against England of 142 for the sixth wicket.
The Black Caps are still 115 runs behind England's first innings total with just four wickets in hand.
Despite having scored more than 12,000 test runs at an average of 46.02, the 33-year-old Cook has appeared woefully out of form since England arrived Down Under for their dual tours of Australia and New Zealand. De Grandhomme (72) signalled his intention with three boundaries in the 21st over off Mark Wood (0-50).
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De Grandhomme was 53 not out at tea, Watling 40 and New Zealand trailed England by 184 runs.
He drew Latham onto the front foot, finding the outside edge as the opener looked to drive, then pushed one out a touch wider to Ross Taylor who couldn't resist a flat-footed flash which sent a catch to Alastair Cook at first slip.
England bowler Stuart Broad, who took 4-38, on Test Match Special: "It's the best I've felt for a while". "New Zealand are 100-something behind and with the new ball around the corner, we've seen the damage the new ball can do on this pitch".
That same newspaper reported that Andrew Strauss, the England director of cricket, has banned England's players from talking in detail about the episode, and Bairstow only commented in the most general terms.
"I get to 20 and think I'm like Bradman". Whether the new-ball talk was a ploy to get the competitive juices flowing or not, it may now be a little while before the Anderson-Broad partnership is broken.
"Watling didn't really give us a chance. I felt strong at the crease, I had good pace, my wrist was in a really good position and I got the ball moving". This time too they picked 10 scalps between them.
"The short-pitched bowling is worth a crack for a couple of overs, but it's been over-done today".