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Package sage :: Package ellcurve :: Module ellcurve :: Class Point |
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Method Summary | |
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__init__(self,
curve,
x,
y,
check)
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Add self to right. | |
__getitem__(self,
n)
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__mul__(self,
m)
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__neg__(self)
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__repr__(self)
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__rmul__(self,
m)
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__sub__(self,
right)
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base_field(self)
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base_ring(self)
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curve(self)
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The height of the point. | |
is_infinity(self)
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Return the order of this point on the elliptic curve. | |
Returns the p-adic cyclotomic height of the point. | |
sigma(self,
p)
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x(self)
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xy(self)
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y(self)
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Method Details |
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__add__(self,
right)
Add self to right.
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height(self)The height of the point. |
order(self)Return the order of this point on the elliptic curve. If the point has infinite order, returns 0. EXAMPLE:>>> E = EllipticCurve([0,0,1,-1,0]) >>> P = E([0,0]); P (0, 0) >>> P.order() 0 >>> E = EllipticCurve([0,1]) >>> P=E([-1,0]) >>> P.order() 2 |
padic_height(self, p)Returns the p-adic cyclotomic height of the point. padic_height(p) Input: p: a prime number Output: p-adic cyclotomic height, which is a single p-adic number. Algorithm: We compute this height using the following formula, which is valid for points that are in the intersection of the identity component of the Neron model with the kernel of reduction modulo p: h(P) = 1/2 * sum_{ell!=p} sup(0,-ord_ell(x(P))) + log_p(sigma_p(-x(P)/y(P)) / e), where P=(a/e^2, b/e^3) with gcd(a,e)=1, and where the first sum is over primes ell that don't equal p. If P isn't in the subgroup mentioned above, let n be a positive integer so that n*P is in that subgroup. Then we return h(n*P)/(n**2), which does not depend on the choice of n, and is defined using the above formula. |
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