[87][88][89][90][91][92][93][94][95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v(τ ) by computing the affine models of modular curves X(Γ ) with Γ = Γ 1 (8) ∩ Γ 0 (16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(τ ) is a modular unit over Z we give a new proof of the fact that the singular values of v(τ ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields.Video. For a video summary of this paper, please visit http:// www.youtube.com/watch?v=FWdmYvdf5Jg.

We first prove Sun's three conjectures [Z.H. Sun, On the number of incongruent residues of x 4 +ax 2 +bx modulo p, J. Number Theory 119 (2006) 210-241; Z.H. Sun, http://sfb.hytc.edu.cn/xsjl/szh/, 2000, June] on the number of rational points of some elliptic curves over finite fields F p , which are related to the congruence cubic and quartic residue. And we provide some examples and comments concerning these conjectures.