C170 pilot reported a NMAC while turning crosswind in the traffic pattern when another departing aircraft made an early; unannounced crosswind turn; passing below the reporter's aircraft.

Narrative

I'm a private pilot and was flying VFR with an instructor to practice takeoffs and landings. I departed Runway XX at ZZZ after my radio call at XA:03. In the call I stated 'closed traffic'. There was a twin immediately behind me on the taxiway and other traffic in the pattern. At XA:04 the twin Cessna announced departing Runway XX 'left turn out departure'. I announced crosswind at XA:05. Just after completing the turn downwind our Garmin GTN-650 navigator gave a visual and audible traffic alert. We then saw a light twin making a climbing left turn. That aircraft passed beneath us at XA:05 according to the playback of our wing-mounted video camera. We were at 1;820 feet MSL westbound at 78 kts. (100 feet below traffic pattern altitude and still climbing) and the twin was at 1;480 feet MSL northbound at 120 kts when they passed beneath us at approximately* XA:06. I announced downwind at XA:07; approximately mid-field and subsequently landed while the twin departed to the west. Neither pilot made a radio call to the other.