Data analysis for Theorem 7

Data Analysis for Theorem : Number of non-singular elliptic curve over Z_p^m is $\phi(p^{2m})$ , gcd(p^m,6)=1

=> Reduced weierstrass Elliptic curve y² = x³ + ax + b is non-singular iff discriminant $discriminant$ is unit over Z_p^m, gcd(p^m,6)=1
=> To find number of non-singular elliptic curves over Z_p^m,we need to find how many pairs of (a,b) gives discriminant as unit element
so #ways $discriminant$ is unit = (total curves over Z_p^m) - (#ways $discriminant$ is non-unit) = p^2m - (#ways $discriminant$ is non-unit)
where $discriminant$ = $-16(4a^3+27b^2)$ , $a,b\in\mathbb{Z}_{p^m}$

Z₅^m
5
25
125
625
3125
15625

Z₇^m
7
49
343
2401
16807

Z₁₁^m
11
121
1331
14641

Z₁₃^m
13
169
2197
28561

Data Analysis for Theorem : Number of non-singular elliptic curve over Zpm is , gcd(pm,6)=1

Data Analysis for Theorem : Number of non-singular elliptic curve over Z_p^m is $\phi(p^{2m})$ , gcd(p^m,6)=1