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Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The cubic polynomial has discriminant equal to p^2 and has three real roots, one positive and two negative. Here we consider the positive root in the case a = 1 corresponding to the prime p = 13. See A348721 and A348722 for the negative roots.

Let r_0 = 4*cos(2*Pi/13)*cos(3*Pi/13): r_0 is the positive root of the cubic equation x^3 - x^2 - 4*x - 1 = 0. The negative roots are r_1 = - 4*cos(4*Pi/13)*cos(6*Pi/13) = - 0.2738905549... and r_2 = - 4*cos(8*Pi/13)*cos(12*Pi/13) = - 1.3772028539....

The roots of the cubic are permuted by the linear fractional transformation x -> - 1/(1 + x) of order 3:

r_1 = - 1/(1 + r_0); r_2 = - 1/(1 + r_1); r_0 = - 1/(1 + r_2).

The quadratic mapping z -> z^2 - 2*z - 2 also cyclically permutes the roots. The mapping z -> - z^2 + z + 3 gives the inverse cyclic permutation of the roots.

The algebraic number field Q(r_0) is a totally real cubic field with class number 1 and discriminant equal to 13^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. See Shanks, Table 1, entry corresponding to a = 1.

The real numbers r_0 and 1 + r_0 are two independent fundamental units of the field Q(r_0). See Shanks. In Cusick and Schoenfeld, r_0 and r_1 (denoted there by E_1 and E_2) are taken as a fundamental pair of units (see case 4 in the table).

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The polynomial has three real roots, one positive and two negative.

In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.3447123654..., r_1 = - 0.1576115578... and r_2 = - 1.1871008076....

Here we consider the absolute value of the root r_2. See A348726 (r_0) and A348727 (|r_1|) for the other two roots.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

In the case a = 1, corresponding to the prime p = 13, Shanks' cyclic cubic is x^3 - x^2 - 4*x - 1 of discriminant 13^2. The three real roots of the cubic are r_0 = 4*cos(2*Pi/13)*cos(3*Pi/13) = 2.6510934089..., r_1 = - 4*cos(4*Pi/13)*cos(6*Pi/13) = - 0.2738905549... and r_2 = - 4*cos(8*Pi/13)*cos(12*Pi/13) = - 1.3772028539.... Here we consider the absolute value of the root r_1.

See A348720 and A348722 for the other two roots.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

In the case a = 1, corresponding to the prime p = 13, Shanks' cyclic cubic is x^3 - x^2 - 4*x - 1 of discriminant 13^2. The three real roots of the cubic are r_0 = 4*cos(2*Pi/13)*cos(3*Pi/13) = 2.6510934089..., r_1 = - 4*cos(4*Pi/13)*cos(6*Pi/13) = - 0.2738905549... and r_2 = - 4*cos(8*Pi/13)*cos(12*Pi/13) = - 1.3772028539.... Here we consider the absolute value of the root r_2.

See A348720 and A348721 for the other two roots.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348724 and A348725.

The quadratic mapping z -> z^2 - 3*z - 2 cyclically permutes the roots of the cubic: the mapping z -> - z^2 + 2*z + 4 gives the inverse cyclic permutation of the three roots.

The algebraic number field Q(r_0) is a totally real cubic field of class number 1 and discriminant equal to 19^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. The real numbers r_0 and 1 + r_0 are two independent fundamental units of the field Q(r_0). See Shanks. In Cusick and Schoenfeld, r_0 and r_1 (denoted there by E_1 and E_2) are taken as a fundamental pair of units (see case 9 in the table).

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348723 and A348725.

Here we consider the absolute value of the root r_1. In Cusick and Schoenfeld r_1 is denoted by E_2. See case 9 in the table.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348723 (r_0) and A348724 (|r_1|).

Here we consider the absolute value of the root r_2.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The polynomial has three real roots, one positive and two negative.

In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.3447123654..., r_1 = - 0.1576115578... and r_2 = - 1.1871008076.... Here we consider the positive root r_0. See A348727 (|r_1|) and A348728 (|r_2|) for the other two roots.

The algebraic number field Q(r_0) is a totally real cubic field with class number 1 and discriminant equal to 37^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. The real numbers r_0 and 1 + r_0 are two independent fundamental units of the field Q(r_0). See Shanks. In Cusick and Schoenfeld, r_0 and r_1 (denoted there by E_1 and E_2) are taken as a fundamental pair of units (see case 37 in the table).

Let R = <1, 6, 8, 10, 11, 14, 23, 26, 27, 29, 31, 36> denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_37, with cosets 2*R = {2, 9, 12, 15, 16, 17, 20, 21, 22, 25, 28, 35} and 3*R = {3, 4, 5, 7, 13, 18, 19, 24, 30, 32, 33, 34}.

Define R(k) = sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*sin(10*k*Pi/37)* sin(11*k*Pi/37)*sin(14*k*Pi/37). Then the three roots of the cubic x^3 - 4*x^2 - 7*x - 1 are

r_0 = - R(2)/R(3) = 5.3447123654..., r_1 = - R(1)/R(2) = - 0.1576115578... and r_2 = R(3)/R(1) = - 1.1871008076....

The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots of the cubic polynomial.

The quadratic mapping z -> z^2 - 5*z - 2 also cyclically permutes the roots of the cubic: the inverse cyclic permutation of the roots is given by z -> - z^2 + 4*z + 6.

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The polynomial has three real roots, one positive and two negative.

In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.344712365..., r_1 = - 0.1576115578... and r_2 = - 1.187100807.... Here we consider the absolute value of the root r_1 (|E_2| in the notation of Cusick and Schoenfeld). See A348726 (r_0) and A348728 (|r_2|) for the other two roots.