We study a class of strictly weakly integral fusion categories I_{N, ζ}, where N≥1 is a natural number and ζ is a 2^Nth root of unity, that we call N-Ising fusion categories. An N-Ising fusion category has Frobenius-Perron dimension 2^N+1 and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order Z_2^N. We show that every braided N-Ising fusion category is prime and also that there exists a slightly degenerate N-Ising braided fusion category for all N > 2. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided N-Ising fusion categories.

J. Dong is partially supported by the Natural Science Foundation of Jiangsu Providence (Grant No. BK20201390), the startup foundation for introducing talent of NUIST (Grant No. 2018R039), and the Natural Science Foundation of China (Grant No. 11201231). S. Natale is partially supported by CONICET and Secyt-UNC. The work of S. Natale was done in part during visits to NUIST in Nanjing, and ECNU in Shanghai; she thanks both mathematics departments for the outstanding hospitality.