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A060838
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Rank of elliptic curve x^3 + y^3 = n.
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18
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0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1
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OFFSET
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1,19
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COMMENTS
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The elliptic curve X^3 + Y^3 = D*Z^3 where D is a rational integer has a birationally equivalent form y^2*z = x^3 - 2^4*3^3*D^2*z^3 where x = 2^2*3*D*Z, y = 2^2*3^3*D*(Y - X), z = X + Y (see p. 123 of Stephens). Taking z = 1 and 2^2*3^3 = 432 yields y^2 = x^3 - 432*D^2, which is the Weierstrass form of the elliptic curve used by John Voight in the Magma program below. - Ralf Steiner, Nov 11 2017
Zagier and Kramarz studied the analytic rank of the curve E: x^3 + y^3 = m, where m is cubefree. They computed L(E,1) for 0 < m <= 70000 and also L'(E,1) if the sign of the functional equation for L(E,1) was negative. In the second case the range was only 0 < m <= 20000. - Attila Pethő, Posting to the Number Theory List, Nov 11 2017
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LINKS
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...
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PROG
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(Magma)
seq := [];
M := 10000;
for m := 1 to M do
E := EllipticCurve([0, -432*m^2]);
Append(~seq, Rank(E));
end for;
seq;
// John Voight, Nov 02 2017
(PARI) {a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, -432*n^2]))[1]} \\ Seiichi Manyama, Aug 25 2019
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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Noam Katz (noamkj(AT)hotmail.com), May 02 2001
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EXTENSIONS
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STATUS
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approved
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