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Search: a060748 -id:a060748
Displaying 1-10 of 15 results found.
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A060838
Rank of elliptic curve x^3 + y^3 = n.
+10
18
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,19
COMMENTS
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
LINKS
John Voight and Joseph L. Wetherell, Table of n, a(n) for n = 1..10000
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1<=n<=200 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 201<=n<=500 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 501<=n<=1000 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1001<=n<=1500 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1501<=n<=2000 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 2001<=n<=2500 (text in Japanese)
...
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 8501<=n<=9000 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9001<=n<=9500 (text in Japanese)
Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9501<=n<=10000 (text in Japanese)
N. M. Stephens, The Diophantine equation X^3 + Y^3 = D Z^3 and the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 231 (1968), 121-162.
D. Zagier and G. Kramarz, Numerical investigations related to the L-series of certain elliptic curves, J. Indian Math. Soc. 52 (1987), 51-60 (the Ramanujan Centenary volume).
PROG
(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
CROSSREFS
Cf. A060748 (positions of records in this sequence), A060950.
KEYWORD
nonn,nice
AUTHOR
Noam Katz (noamkj(AT)hotmail.com), May 02 2001
EXTENSIONS
Many thanks to Andrew V. Sutherland, John Voight, and Joseph L. Wetherell, who all responded to my request for additional terms for this sequence. - N. J. A. Sloane, Nov 01 2017
STATUS
approved
A060953
Rank of elliptic curve y^2 = x^3 + n*x.
+10
15
0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,14
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1000
H. Mishima, Tables of Elliptic Curves
F. Richman, Elliptic curves
K. Rubin and A. Silverberg, Ranks of elliptic curves
FORMULA
a(-n) = A060952(n). - Michael Somos, Dec 15 2011
PROG
(PARI) { A060953(n) = ellanalyticrank( ellinit([0, 0, 0, n, 0]) )[1]; }
CROSSREFS
Cf. A060748, A060838, A060950-A060953.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, May 10 2001
EXTENSIONS
Lambert Klasen (Lambert.Klasen(AT)gmx.net), Mar 31 2005, kindly rechecked this sequence against the Mishima web site and found no errors.
Corrected Apr 10 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator.
Extended by Max Alekseyev, Mar 09 2009
STATUS
approved
A002155
Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 2.
(Formerly M4957 N2125)
+10
14
15, 17, 24, 37, 43, 57, 63, 65, 73, 79, 89, 101, 106, 122, 129, 131, 142, 145, 148, 151, 161, 164, 168, 171, 186, 195, 197, 198, 204, 217, 222, 223, 225, 229, 232, 233, 248, 252, 260, 265, 268, 269, 281, 294, 295, 297, 303, 322, 331, 337, 347, 350, 353, 360, 366, 369, 373, 377, 381, 388, 389, 392, 404, 409, 412, 414, 433, 449, 464, 469, 481, 483, 485, 492
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1724 (using Gebel)
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
L. Lehman, Elliptic Curves of Rank Two. [broken link]
H. Mishima, Tables of Elliptic Curves.
PROG
(Magma) for k in[1..500] do if Rank(EllipticCurve([0, 0, 0, 0, k])) eq 2 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019
CROSSREFS
Cf. A060950, A002151, A002153, A102833, A060748, A060838, A060951-A060953.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from James R. Buddenhagen, Feb 18 2005
STATUS
approved
A060950
Rank of elliptic curve y^2 = x^3 + n.
+10
14
0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,15
COMMENTS
The curves for n and -27*n are isogenous (as Noam Elkies points out--see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves
T. Womack, Minimal-known positive and negative k for Mordell curves of given rank
FORMULA
a(n) = A060951(27*n) and A060951(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013
EXAMPLE
a(1) = A060951(27) = a(729) = 0. - Jonathan Sondow, Sep 10 2013
PROG
(PARI) a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, n]))[1] \\ Jianing Song, Aug 24 2022
(PARI) apply( {A060950(n)=ellrank(ellinit([0, n]))[1]}, [1..99]) \\ For PARI version < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024
CROSSREFS
Cf. A031507, A002151, A002153, A002155, A102833, A179124.
Cf. A060748, A060838, A060951, A060952, A060953.
Cf. A081119 (number of integral solutions to Mordell's equation y^2 = x^3 + n).
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, May 10 2001
EXTENSIONS
Corrected by James R. Buddenhagen, Feb 18 2005
STATUS
approved
A002151
Numbers k for which rank of the elliptic curve y^2 = x^3 + k is 0.
(Formerly M3271 N1321)
+10
13
1, 4, 6, 7, 13, 14, 16, 20, 21, 23, 25, 27, 29, 32, 34, 42, 45, 49, 51, 53, 59, 60, 64, 70, 75, 78, 81, 84, 85, 86, 87, 88, 90, 93, 95, 96, 104, 109, 114, 115, 116, 123, 124, 125, 135, 137, 140, 144, 153, 157, 158, 159, 160, 162, 165, 167, 173, 175, 176, 178
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..2907 (using Gebel)
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
L. Lehman, Elliptic Curves of Rank Zero.
H. Mishima, Tables of Elliptic Curves.
PROG
(Magma) for k in[1..200] do if Rank(EllipticCurve([0, 0, 0, 0, k])) eq 0 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019
CROSSREFS
Cf. A060950, A002153, A002155, A102833, A060748, A060838, A060951, A060952, A060953.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
Corrected and extended by James R. Buddenhagen, Feb 18 2005
The missing entry 123 was added by T. D. Noe, Jul 24 2007
STATUS
approved
A002153
Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 1.
(Formerly M0682 N0251)
+10
12
2, 3, 5, 8, 9, 10, 11, 12, 18, 19, 22, 26, 28, 30, 31, 33, 35, 36, 38, 39, 40, 41, 44, 46, 47, 48, 50, 52, 54, 55, 56, 58, 61, 62, 66, 67, 68, 69, 71, 72, 74, 76, 77, 80, 82, 83, 91, 92, 94, 97, 98, 99, 100, 102, 103, 105, 107, 108, 110, 111, 112, 117, 118, 119
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..5111 (using Gebel).
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
H. Mishima, Tables of Elliptic Curves.
PROG
(Magma) for k in[1..200] do if Rank(EllipticCurve([0, 0, 0, 0, k])) eq 1 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019
CROSSREFS
Cf. A060950, A002151, A002155, A102833, A060748, A060838, A060951-A060953.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
Corrected and extended by James R. Buddenhagen, Feb 18 2005
STATUS
approved
A060951
Rank of elliptic curve y^2 = x^3 - n.
+10
12
0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,11
COMMENTS
The curves for n and -27*n are isogenous (as Noam Elkies points out--see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves
T. Womack, Minimal-known positive and negative k for Mordell curves of given rank
FORMULA
a(n) = A060950(27*n) and A060950(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013
EXAMPLE
a(1) = A060950(27) = a(729) = 0. - Jonathan Sondow, Sep 10 2013
PROG
(PARI) {a(n) = if( n<1, 0, length( ellgenerators( ellinit( [ 0, 0, 0, 0, -n], 1))))} /* Michael Somos, Mar 17 2011 */
(PARI) apply( {A060951(n)=ellrank(ellinit([0, -n]))[1]}, [1..99]) \\ For version < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024
CROSSREFS
Cf. A031508, A002150, A002152, A002154, A179136, A179137.
Cf. A060748, A060838, A060950, A060952, A060953.
Cf. A081120 (number of integral solutions to Mordell's equation y^2 = x^3 - n).
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, May 10 2001
EXTENSIONS
Corrected Apr 08 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator.
STATUS
approved
A060952
Rank of elliptic curve y^2 = x^3 - n*x.
+10
10
0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,17
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1000
H. Mishima, Tables of Elliptic Curves
PROG
(PARI) {a(n) = ellanalyticrank(ellinit([0, 0, 0, -n, 0]))[1]} \\ Seiichi Manyama, Sep 16 2018
CROSSREFS
Cf. A060748, A060838, A060950-A060953.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, May 10 2001
EXTENSIONS
Corrected Apr 08 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator and each curve of rank three thrice.
STATUS
approved
A102833
Numbers n for which rank of the elliptic curve y^2=x^3+n is 3.
+10
9
113, 141, 316, 346, 359, 427, 443, 506, 537, 568, 659, 681, 730, 745, 873, 892, 899, 940, 997, 1016, 1025, 1090, 1149, 1157, 1171, 1213, 1304, 1305, 1342, 1367, 1373, 1478, 1522, 1639, 1646, 1737, 1753, 1772, 1811, 1841, 1897, 1907, 1954, 2024, 2143
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n=1..250 (using Gebel).
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves
PROG
(Magma) for k in[1..2000] do if Rank(EllipticCurve([0, 0, 0, 0, k])) eq 3 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019
CROSSREFS
Cf. A060950, A002151, A002153, A002155, A060748, A060838, A060951-A060953.
KEYWORD
nonn
AUTHOR
James R. Buddenhagen, Feb 18 2005. Entry revised by N. J. A. Sloane, Jun 10 2012
EXTENSIONS
More terms from T. D. Noe, Jul 24 2007
STATUS
approved
A309960
Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.
+10
6
1, 2, 3, 4, 5, 8, 10, 11, 14, 16, 18, 21, 23, 24, 25, 27, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 54, 55, 57, 59, 60, 64, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 125, 128, 129, 131, 135, 137
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
A060838(a(n)) = 0.
PROG
(PARI) for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==0, print1(k", ")))
(PARI) is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E, eri, mwr, ar); if(r<6, return(1)); E=ellinit([0, 16*r^2]); eri=ellrankinit(E); mwr=ellrank(eri); if(mwr[1], return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(!ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>0, return(0), mwr[2]<1, return(1))); "unknown (0 under BSD conjecture)" \\ Charles R Greathouse IV, Jan 24 2023
CROSSREFS
Complement of A159843 \ A000578.
Cf. A060748, A060838, A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 25 2019
STATUS
approved
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