Prime Numbers

360 Chapter 7 ELLIPTIC CURVE ARITHMETIC
arising in the theory of invariants and modular forms [Cohen 2000], [Atkin and
Morain 1993b]. (It is interesting that ∆(q) has the alternative and beautiful
representation q
n≥1 (1 − qn ) 24 , but we shall not use this in what follows.
The first given expression for ∆(q) is more amenable to calculation since the
exponents grow quadratically.)
Algorithm 7.5.8 (Class number and Hilbert class polynomial).
Given a (negative) fundamental discriminant D, this algorithm returns any desired
combination of the class number h(D), the Hilbert class polynomial T ∈ Z[X]
(whose degree is h(D)), and the set of reduced forms (a, b, c) of discriminant D
(whose cardinality is h(D)).
1. [Initialize]
T =1;
b = D mod 2;
r = ⌊ |D|/3⌋;
h =0; //Zeroclasscount.
red = {}; // Empty set of primitive reduced forms.
2. [Outer loop on b]
while(b ≤ r) {
m =(b 2 − D)/4;
for(1 ≤ a and a 2 ≤ m) {
if(m mod a = 0) continue; // Continue ‘for’ loop to force a|m.
c = m/a;
if(b >a) continue; // Continue ‘for’ loop.
3. [Optional polynomial setup]
τ =(−b + i |D|)/(2a); // Note precision (see text following).
f =∆(e 4πiτ )/∆(e 2πiτ ); // Note precision.
j = (256f +1) 3 /f; // Note precision.
4. [Begin divisors test]
if(b == a or c == a or b == 0) {
T = T ∗ (X − j);
h = h +1; // Class count.
red = red ∪ (a, b, c); // New form.
} else {
T = T ∗ (X 2 − 2Re(j)X + |j| 2 );
h = h +2; // Class count.
red = red ∪ (a, ±b, c); // Two new forms.
}
}
}
5. [Return values of interest]
return (combination of) h, round(Re(T (x))),red;
This algorithm is straightforward in every respect except on the issue of
floating-point precision. Note that the function ∆ must be evaluated for