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In this paper, we studied the elliptic curve  y 2  = ( x  –  a )( x 2  +  ax  +  r ),  a ,  r  ∈  Z +  in the case of  a  = 6, r  =  7  which has no relevant conclusions up to now. And we obtained the conclusion of the integer points on Elliptic  y 2  = ( x  – 6)( x 2  + 6 x  +  7 ), elliptic curve in title has only one integer point ( x ,  y ) = (6,0) were proved with the help of the elementary methods such as congruence and Legendre symbol.

The positive integral points on the elliptic curve y 2 = (x + 4)(x 2 − 4x + 5)