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fix docstring format for EllipticCurve_with_prime_order
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src/sage/schemes/elliptic_curves/ell_finite_field.py

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

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