@@ -2962,22 +2962,22 @@ def EllipticCurve_with_prime_order(N):
29622962 a suitable `D` *incrementally*, by enlarging the table `S` by `log(N)`-size
29632963 interval of primes `p` and testing all products of distinct primes `p` (or
29642964 rather `p^*`). We find this difficult to implement without testing
2965- duplicate `D`s, so we instead enlarge the table one prime at a time
2965+ duplicate `D`\ s, so we instead enlarge the table one prime at a time
29662966 (effectively replacing `[r\log(N), (r + 1)\log(N)]` in the paper by `[r,
29672967 r]`). To compensate for the speed loss, we begin the algorithm by
29682968 prefilling `S` with the primes below `1000` (satisfying quadratic
29692969 reciprocity properties). The constant `1000` is determined experimentally
29702970 to be fast for many purposes, and for most `N` we tested we are able to
29712971 find a suitable small `D` without increasing the size of `S`.
29722972
2973- The paper also doesn't specify how to enumerate such `D`s, which recall
2973+ The paper also doesn't specify how to enumerate such `D`\ s, which recall
29742974 should be product of distinct values in the table `S`. We implement this
29752975 with a priority queue (min heap), which also allows us to search for the
2976- suitable `D`s in increasing (absolute value) order. This is suitable for
2976+ suitable `D`\ s in increasing (absolute value) order. This is suitable for
29772977 the algorithm because smaller `D` means the Hilbert class polynomial is
29782978 computed quicker.
29792979
2980- Finally, to avoid repeatedly testing the same `D`s, we require the latest
2980+ Finally, to avoid repeatedly testing the same `D`\ s, we require the latest
29812981 prime to be added to the table to be included as a factor of `D` (see code
29822982 for more explanation). As we need to find integers `x, y` such that `x^2 +
29832983 (-D)y^2 = 4N` with `D < 0` and `N` prime, we actually need `|D| \leq 4N`,
@@ -3022,7 +3022,7 @@ def EllipticCurve_with_prime_order(N):
30223022 sage: E.has_order(N)
30233023 True
30243024
3025- ::
3025+ Another example for large primes ::
30263026
30273027 sage: N = next_prime(2^256)
30283028 sage: E = next(EllipticCurve_with_prime_order(N)); E # random
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