Sorry I don't know where the frame number, if any, is on that bike. Common places are under the bottom bracket and on e-bikes, on the battery support bracket area.
If there's no number, you could get a set of cheap metal number stamps from somewhere like Machine Mart and stamp one yourself on the tubes below the bottom bracket. Alternatively, check with your police whether they operate the "smart liquid" or similar ID scheme. These invisibly plaster your bike with thousands of tiny dots, each bearing a number unique to you, enabling the police to subsequently ID the bike. Application is usually free.
The little box is the Hall effect motor's controller. Hall effect motors are different from other types in that they are not complete in themselves. They can only work if they have a controller to communicate with, that controller therefore being a remotely situated part of the motor.
The motors work with Hall sensors which magnetically check the position of the motor's rotor. That information is sent to the controller which then decides which of the motor's phase coils to send a pulse of current to in order for it to turn in the correct direction. There's a constant stream of positional information then fed and power pulses returned to keep the motor running.
The throttle or pedelec control is in charge of the controller in turn, instructing it when to start and how much power to issue.
There are three wire motors which have to be pedalled off the mark before the motor will run, but most have eight wires, three for feeding power to the phase coils, and five thin ones for the positional sensing information, all connected to the controller at one end and going into the hub motor at the other.
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