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Fix whitespace characters
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1 changed files with 4 additions and 4 deletions
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@ -75,19 +75,19 @@ to be square), the curve is *complete*. In this case the
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four-torsion subgroup is cyclic, and we
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can write it explicitly as
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$$
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\mathcal E\_{a,d}[4] = \\{ (0,1),\; (1/\sqrt a, 0),\; (0, -1),\; (-1/\sqrt{a}, 0)\\}.
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\mathcal E\_{a,d}[4] = \\{ (0,1),\\; (1/\sqrt a, 0),\\; (0, -1),\\; (-1/\sqrt{a}, 0)\\}.
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$$
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These are the only points with \\(xy = 0\\); the points with \\( y
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\neq 0 \\) are \\(2\\)-torsion. The \\(\mathcal
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E\_{a,d}[4]\\)-coset of \\(P = (x,y)\\) is then
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$$
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P + \mathcal E\_{a,d}[4] = \\{ (x,y),\; (y/\sqrt a, -x\sqrt a),\; (-x, -y),\; (-y/\sqrt a, x\sqrt a)\\}.
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P + \mathcal E\_{a,d}[4] = \\{ (x,y),\\; (y/\sqrt a, -x\sqrt a),\\; (-x, -y),\\; (-y/\sqrt a, x\sqrt a)\\}.
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$$
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Notice that if \\(xy \neq 0 \\), then exactly two of
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these points have \\( xy \\) non-negative, and they differ by the
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\\(2\\)-torsion point \\( (0,-1) \\). This means that we can select
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a representative modulo \\(\mathcal
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E\_{a,d}[2] \\) by requiring \\(xy\\) nonnegative and \\(y \neq
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a representative modulo \\(\mathcal E\_{a,d}[2] \\)
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by requiring \\(xy\\) nonnegative and \\(y \neq
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0\\), and we can ensure this condition by conditionally adding a
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\\(4\\)-torsion point if \\(xy\\) is negative or \\(y = 0\\).
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