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In this paper, all spaces are localized at the prime two. Let  T (1) be the Ravenel spectrum characterized by the  BP  ∗ -homology as  BP  ∗ [ t  1 ],  T (1)/( v  1 ) be the cofiber of the self map  v  1  : Σ 2  T (1) →  T (1) and  L  2  denote the Bousfield localization functor with respect to  v  2  −1  BP  ∗ . In this paper, we compute the homotopy groups π ∗ ( L  2  T (1)/( v  1 )) by determining the  E  ∞ -term of its Adams-Novikov spectral sequence (ANSS). From the  E  2 -term of the ANSS for π ∗ ( L  2  T (1)/( v  1 )), we determine a subgroup of the  E  2 -term for π ∗ ( L  2  T (1)). We also show that the  E  4 -term for π ∗ ( L  2  T (1)) has horizontal vanishing line.

π ∗ ( L 2 T (1)/( v 1 )) and its applications in computing π ∗ ( L 2 T (1)) at the prime two