A014602 -id:A014602 - cp's OEIS frontend

A305416 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 8-group, C_2 x C_2 x C_2.

420, 660, 840, 1092, 1155, 1320, 1380, 1428, 1540, 1848, 1995, 3003, 3315

Offset: 1

Vincenzo Librandi, Jun 12 2018

Intersection of A046005 and A003644. Note that A003644 = A014602 union A014603 union A192322 union {a(n)} union {5460}. - Jianing Song, Jul 12 2018

Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler, Principally Polarized Squares of Elliptic Curves with Field of Moduli Equal To Q, arXiv:1806.03826 [math.NT], 2018 (see table 1 page 4).

Cf. A003644, A014602, A014603, A192322.

Subsequence of A046005 and A003644.

A046005 = Import["https://oeis.org/A046005/b046005.txt", "Table"][[All, 2]];
A003644 = Import["https://oeis.org/A003644/b003644.txt", "Table"][[All, 2]];
A046005 ~Intersection~ A003644 (* Jean-François Alcover, Sep 25 2019 *)

ok(n)={isfundamental(-n) && [2, 2, 2] == quadclassunit(-n).cyc} \\ Andrew Howroyd, Jul 20 2018

A316743 Discriminants of imaginary fields whose class group has exponent 2, negated.

Original entry on oeis.org

15, 20, 24, 35, 40, 51, 52, 84, 88, 91, 115, 120, 123, 132, 148, 168, 187, 195, 228, 232, 235, 267, 280, 312, 340, 372, 403, 408, 420, 427, 435, 483, 520, 532, 555, 595, 627, 660, 708, 715, 760, 795, 840, 1012, 1092, 1155, 1320, 1380, 1428, 1435, 1540, 1848, 1995, 2280, 3003, 3315, 5460

Offset: 1

Jianing Song, Jul 20 2018

This sequence lists the negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r, r > 0.
These are the negated fundamental discriminants in A133288.
Also numbers in A003644 but not in A014602. Equals A014603 union A192322 union A305416 union {5460}.

Shin-ichi Katayama, Rational Points on the Parabola and the Arithmetic of Related Algebraic Tori, J. Math. Tokushima Univ. (2024) Vol. 58, 11-31. See p. 30.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
P. J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117-124.

Cf. Negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r: A014602 (r=0), A014603 (r=1), A192322 (r=2), A305416 (r=3).

Subsequence of A003644 and A133288.

ok(n)={isfundamental(-n) && quadclassunit(-n).no > 1 && !#select(k->k<>2, quadclassunit(-n).cyc)} \\ Andrew Howroyd, Jul 20 2018

A351666 Discriminants of imaginary quadratic fields with class number 28 (negated).

Original entry on oeis.org

831, 935, 1095, 1311, 1335, 1364, 1455, 1479, 1496, 1623, 1703, 1711, 1855, 1976, 2024, 2055, 2120, 2127, 2324, 2359, 2431, 2455, 2564, 2607, 2616, 2703, 3224, 3272, 3396, 3419, 3487, 3535, 3572, 3576, 3608, 3624, 3731, 3848, 3995, 4040, 4183, 4279, 4344

Offset: 1

Andy Huchala, Feb 16 2022

Sequence contains 457 terms; largest is 126043.

The class groups associated to 174 of the above discriminants are isomorphic to C_28, and the remaining 283 have a class group isomorphic to C_14 X C_2.

Andy Huchala, Table of n, a(n) for n = 1..457
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

isok(n) = {isfundamental(-n) && quadclassunit(-n).no == 28}; \\ Michel Marcus, Mar 02 2022

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 28]

A351679 Discriminants of imaginary quadratic fields with class number 41 (negated).

Original entry on oeis.org

1151, 2551, 2719, 3079, 3319, 3511, 6143, 9319, 9467, 10499, 10903, 11047, 11483, 11719, 11987, 12227, 12611, 13567, 14051, 14411, 14887, 14983, 16067, 16187, 19763, 20407, 20771, 21487, 22651, 24971, 25171, 26891, 26987, 27739, 28547, 29059, 29251, 30859

Offset: 1

Andy Huchala, Mar 28 2022

Sequence contains 109 terms; largest is 296587.

The class group of Q[sqrt(-d)] is isomorphic to C_41 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..109
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 41]

A330162 For imaginary fundamental discriminants -d, define b(-d) to be the smallest prime p such that Kronecker(-d,p) = 1. Sequence gives d such that b(-d)^3 > d/4 > b(-d)^2.

Original entry on oeis.org

23, 31, 56, 59, 68, 83, 104, 107, 136, 139, 184, 211, 219, 244, 259, 264, 276, 283, 291, 292, 307, 328, 331, 339, 355, 376, 379, 388, 411, 424, 436, 451, 456, 472, 499, 523, 547, 552, 568, 580, 628, 643, 667, 712, 723, 763, 772, 787, 808, 820, 835, 843, 852, 868, 883

Offset: 1

Jianing Song, Dec 04 2019

nonn

It seems that this sequence contains 810 terms, the largest being 1154008. In general, it seems that for any t > 0, b(-d) = o(d^t) as -d -> -oo.
For fundamental discriminants -d, we want to determine the size of b(-d), i.e., the size of the smallest prime that decomposes in Q[sqrt(-d)].
Let K = Q[sqrt(-d)], O_K be the ring of integers over K, so O_K is a Dedekind domain. Let E(-d) be the exponent of the ideal class group of O_K (the exponent of a group G is the smallest e > 0 such that x^e = I for all x in G, where I is the group identity).
If Kronecker(-d,p) = 1, it is well-known that p*O_K is the product of two distinct prime ideals of O_K, say, p*O_K = I*I'. By the property of the ideal class group of Q[sqrt(-d)], I^(k*e) must be principal, e = E(-d). Let t*O_K = I^(k*e), then t/p is not an algebraic integer, and the norm of t is p^e. Define f(x,y) = x^2 + (d/4)*y^2 if -d == 0 (mod 4), x^2 + x*y + ((d+1)/4)*y^2 otherwise, it is easy to see f(x,y) = p^(k*e) has integral solutions (x,y) such that gcd(x,y) = 1.
If f(x,y) = p^(k*e) < d, then |y| = 1, so 4*p^(k*e) - d must be a (positive) square. Setting k = 1 gives b(-d) > (d/4)^(1/e) (and furthermore we have: if Kronecker(-d,p) = 1 and p^(k*e) < d, then k = 1, or (p,k,e,d) = (2,2,1,7), (3,2,1,11)).
If E(-d) = 3, then d is in this sequence.
We also have the following observations (not proved):
(a) if e = 2 (i.e., d is in A003644\A014602 = A316743), then b(-d) < d/4;
(b) if e > 2, then b(-d) < sqrt(d/4) (it can be proved by using deeper algebraic number theory that b(-d) < 2*sqrt(d)/Pi).
If these observations are true, this sequence is also the list of d such that b(-d) > (d/4)^(1/3) and d is not in A003644.
Note that 5460 is conjectured to be the largest term in A003644. Therefore, it seems that b(-d) < sqrt(d/4) for all d > 5460; it seems that b(-d) < (d/4)^(1/3) for all d > 1154008.
Among the known terms:
(1) the term d with the largest E(-d) is d = 998328 with E(-d) = 66.
(2) the term d with the largest b(-d) is d = 656755 with b(-d) = 79.
(3) the largest prime is d = 90787 with E(-d) = 23.

			The smallest prime p such that Kronecker(-499,p) = 1 is p = 5, and 5^3 > 499/4 > 5^2, so 499 is a term.

Jianing Song, Table of n, a(n) for n = 1..810

Cf. A014602, A003644, A316743, A306538.

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
isA330162(d) = (d>0) && isfundamental(-d) && b(-d) > sqrtn(d/4,3) && b(-d) < sqrt(d/4)

A351665 Discriminants of imaginary quadratic fields with class number 27 (negated).

Original entry on oeis.org

983, 1231, 1399, 1607, 1759, 1879, 1999, 3271, 3299, 3943, 4903, 6007, 6011, 7699, 8867, 10531, 10939, 11003, 11027, 11383, 11491, 11779, 11939, 13411, 14243, 14723, 15107, 15739, 16411, 16547, 17443, 17627, 17659, 17747, 18587, 18787, 18859, 19051, 19427

Offset: 1

Andy Huchala, Feb 16 2022

Sequence contains 93 terms; largest is 103387.

The class group of Q[sqrt(-d)] is isomorphic to C_9 X C_3 for d = 3299, 19427, 34603, 89923, and 98443. For all other d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_27.

Andy Huchala, Table of n, a(n) for n = 1..93
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

isok(n) = {isfundamental(-n) && quadclassunit(-n).no == 27}; \\ Michel Marcus, Mar 02 2022

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 27]

A351667 Discriminants of imaginary quadratic fields with class number 29 (negated).

Original entry on oeis.org

887, 2287, 2311, 2383, 2939, 3583, 3659, 3823, 4451, 4519, 5051, 5743, 6947, 7207, 7643, 7687, 8863, 8963, 9323, 12323, 13763, 13883, 14387, 15139, 15227, 15443, 15467, 15859, 16427, 17491, 20483, 20507, 22051, 23059, 23251, 24859, 25523, 28403, 29587, 29723

Offset: 1

Andy Huchala, Mar 24 2022

Sequence contains 83 terms; largest is 166147.

The class group of Q[sqrt(-d)] is isomorphic to C_29 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..83
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351666.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 29]

A351668 Discriminants of imaginary quadratic fields with class number 30 (negated).

Original entry on oeis.org

671, 815, 1007, 1844, 2036, 2071, 2191, 2264, 2319, 2599, 2708, 3188, 3223, 3284, 3439, 3991, 4087, 4276, 4696, 4835, 4859, 4979, 5579, 5912, 6107, 6459, 6463, 6488, 6535, 6635, 7087, 7115, 7303, 7576, 7835, 7971, 8259, 8267, 8367, 8483, 8948, 9019, 9076

Offset: 1

Andy Huchala, Mar 24 2022

Sequence contains 255 terms; largest is 134467.

The class group of Q[sqrt(-d)] is isomorphic to C_30 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..255
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 30]

A351669 Discriminants of imaginary quadratic fields with class number 31 (negated).

Original entry on oeis.org

719, 911, 2927, 3251, 3727, 3779, 4159, 4951, 5651, 6131, 6491, 7639, 8647, 9203, 10427, 11863, 12347, 12923, 13043, 13219, 13687, 14627, 14731, 15923, 17987, 18803, 19219, 20611, 24691, 24979, 28051, 32083, 32363, 35491, 38851, 39667, 39883, 41227, 41539

Offset: 1

Andy Huchala, Mar 24 2022

Sequence contains 73 terms; largest is 133387.

The class group of Q[sqrt(-d)] is isomorphic to C_31 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..73
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 31]

A351670 Discriminants of imaginary quadratic fields with class number 32 (negated).

Original entry on oeis.org

791, 1119, 1239, 1463, 1551, 1767, 1784, 1943, 2084, 2180, 2276, 2343, 2840, 2847, 2996, 3080, 3156, 3199, 3207, 3236, 3247, 3295, 3428, 3476, 3679, 3812, 3895, 4088, 4296, 4340, 4495, 4584, 4647, 4767, 4868, 4884, 4964, 4980, 4996, 5012, 5064, 5192, 5215

Offset: 1

Andy Huchala, Mar 24 2022

Sequence contains 708 terms; largest is 164803.

The class groups associated to 187 of the above discriminants are isomorphic to C_32, 273 have a class group isomorphic to C_16 X C_2, 160 isomorphic to C_8 X C_2 X C_2, 60 have a class group isomorphic to C_8 X C_4, 15 have a class group isomorphic to C_4 X C_2 X C_2 X C_2, and the remaining 13 have a class group isomorphic to C_4 X C_4 X C_2.

Andy Huchala, Table of n, a(n) for n = 1..708
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 32]

cp's OEIS Frontend

A305416 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 8-group, C_2 x C_2 x C_2.

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A316743 Discriminants of imaginary fields whose class group has exponent 2, negated.

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A351666 Discriminants of imaginary quadratic fields with class number 28 (negated).

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A351679 Discriminants of imaginary quadratic fields with class number 41 (negated).

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A330162 For imaginary fundamental discriminants -d, define b(-d) to be the smallest prime p such that Kronecker(-d,p) = 1. Sequence gives d such that b(-d)^3 > d/4 > b(-d)^2.

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A351665 Discriminants of imaginary quadratic fields with class number 27 (negated).

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A351667 Discriminants of imaginary quadratic fields with class number 29 (negated).

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A351668 Discriminants of imaginary quadratic fields with class number 30 (negated).

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A351669 Discriminants of imaginary quadratic fields with class number 31 (negated).

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