Prime Numbers

7.5 Counting points on elliptic curves 361
complex q arguments. The theory shows that sufficient precision for the whole
algorithm is essentially
δ = π |D| 1
ln 10 a
decimal digits, where the sum is over all primitive reduced forms (a, b, c) of
discriminant D [Atkin and Morain 1993b]. This means that a little more than
δ digits (perhaps δ + 10, as in [Cohen 2000]) should be used for the [Optional
polynomial setup] phase, the ultimate idea being that the polynomial T (x)—
consisting of possibly some linear factors and some quadratic factors—
should have integer coefficients. Thus the final polynomial output in the
form round(Re(T (x))) means that T is to be expanded, with the coefficients
rounded so that T ∈ Z[X]. Algorithm 7.5.8 can, of course, be used in a
multiple-pass
fashion: First calculate just the reduced forms, to estimate
1/a and thus the required precision, then start over and this time calculate
the actual Hilbert class polynomial. In any event, the quantity 1/a is always
O ln 2 |D| .
For reader convenience, we give here some explicit polynomial examples
from the algorithm, where TD refers to the Hilbert class polynomial for
discriminant D:
T−3 = X,
T−4 = X − 1728,
T−15 = X 2 + 191025X − 121287375,
T−23 = X 3 + 3491750X 2 − 5151296875X + 12771880859375.
One notes that the polynomial degrees are consistent with the class numbers
below. There are further interesting aspects of these polynomials. One is that
the constant coefficient is always a cube. Also, the coefficients of TD grow
radically as one works through lists of discriminants. But one can use in
the Atkin-Morain approach less unwieldy polynomials—the Weber variety—
at the cost of some complications for special cases. These and many more
optimizations are discussed in [Morain 1990], [Atkin and Morain 1993b].
In the Atkin–Morain order-finding scheme, it will be useful to think of
discriminants ordered by their class numbers, this ordering being essentially
one of increasing complexity. As simple runs of Algorithm 7.5.8 would show
(without the polynomial option, say),
h(D) =1forD = −3, −4, −7, −8, −11, −19, −43, −67, −163;
h(D) =2forD = −15, −20, −24, −35, −40, −51, −52, −88, −91, −115,
−123, −148, −187, −232, −235, −267, −403, −427;
h(D) =3forD = −23, −31, −59,... .
That the discriminant lists for h(D) =1, 2 are in fact complete as given here
is a profound result of the theory [Cox 1989]. We currently have complete
lists for h(D) ≤ 16, see [Watkins 2000], and it is known, in principle at least,