Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(952\)\(\medspace = 2^{3} \cdot 7 \cdot 17 \) |
Artin stem field: | Galois closure of 12.0.15192372849934336.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.952.6t1.a.b |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.6664.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 6 x^{11} + 25 x^{10} - 52 x^{9} + 65 x^{8} + 14 x^{7} - 147 x^{6} + 266 x^{5} - 141 x^{4} + \cdots + 59 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{6} + 19x^{3} + 16x^{2} + 8x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a^{3} + 5 a^{2} + 28 a + 25 + \left(22 a^{5} + 23 a^{4} + a^{3} + 16 a^{2} + 22 a + 17\right)\cdot 31 + \left(22 a^{5} + 2 a^{4} + 27 a^{3} + 24 a^{2} + 23 a + 20\right)\cdot 31^{2} + \left(6 a^{5} + 29 a^{4} + 23 a^{3} + a^{2} + a + 7\right)\cdot 31^{3} + \left(23 a^{5} + 16 a^{4} + 16 a^{3} + 15 a^{2} + 25 a + 9\right)\cdot 31^{4} + \left(22 a^{5} + 21 a^{4} + 6 a^{3} + 3 a^{2} + 16 a + 9\right)\cdot 31^{5} + \left(6 a^{5} + 4 a^{4} + 12 a^{3} + 17 a^{2} + 22 a + 5\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 2 }$ | $=$ | \( 11 a^{4} + 29 a^{3} + 10 a^{2} + 15 a + 18 + \left(5 a^{5} + 12 a^{4} + 27 a^{3} + 13 a^{2} + 14 a + 7\right)\cdot 31 + \left(21 a^{5} + 9 a^{4} + 27 a^{3} + 27 a^{2} + 9 a + 22\right)\cdot 31^{2} + \left(14 a^{5} + 3 a^{4} + 27 a^{3} + 22 a^{2} + 22 a + 4\right)\cdot 31^{3} + \left(25 a^{5} + 30 a^{4} + 4 a^{3} + 7 a^{2} + 20 a + 26\right)\cdot 31^{4} + \left(19 a^{5} + 16 a^{4} + 17 a^{3} + 7 a^{2} + 28 a + 10\right)\cdot 31^{5} + \left(30 a^{5} + 18 a^{4} + 7 a^{3} + 26 a^{2} + 7 a + 25\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{5} + 12 a^{4} + 11 a^{3} + 22 a^{2} + 29 a + 12 + \left(23 a^{5} + 20 a^{4} + 26 a^{3} + 29 a^{2} + 8 a + 16\right)\cdot 31 + \left(27 a^{5} + 5 a^{4} + 3 a^{3} + 19 a^{2} + 28 a + 6\right)\cdot 31^{2} + \left(24 a^{5} + 14 a^{4} + 3 a^{3} + 22 a^{2} + a + 22\right)\cdot 31^{3} + \left(27 a^{5} + 17 a^{4} + 8 a^{3} + 9 a^{2} + 7 a + 24\right)\cdot 31^{4} + \left(7 a^{5} + a^{4} + 14 a^{3} + 12 a^{2} + 29 a + 8\right)\cdot 31^{5} + \left(24 a^{5} + 2 a^{4} + 12 a^{3} + a^{2} + 12 a + 1\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{5} + 15 a^{4} + 2 a^{3} + 26 a^{2} + 29 a + 17 + \left(a^{5} + 29 a^{4} + 17 a^{3} + 26 a + 26\right)\cdot 31 + \left(14 a^{5} + 27 a^{4} + 25 a^{3} + 30 a^{2} + 23 a + 10\right)\cdot 31^{2} + \left(4 a^{5} + 15 a^{4} + 13 a^{3} + 29 a^{2} + 12 a + 4\right)\cdot 31^{3} + \left(3 a^{5} + 4 a^{4} + 27 a^{3} + 20 a^{2} + 29 a + 25\right)\cdot 31^{4} + \left(22 a^{5} + 8 a^{4} + 9 a^{3} + 6 a^{2} + 5 a + 17\right)\cdot 31^{5} + \left(17 a^{5} + 29 a^{4} + 26 a^{3} + 12 a^{2} + 11 a + 9\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 5 }$ | $=$ | \( 18 a^{5} + 11 a^{4} + 3 a^{3} + 7 a^{2} + 25 a + 1 + \left(21 a^{5} + 3 a^{4} + 19 a^{3} + 10 a^{2} + a + 9\right)\cdot 31 + \left(11 a^{5} + 3 a^{4} + 16 a^{3} + 30 a^{2} + 27 a + 14\right)\cdot 31^{2} + \left(10 a^{5} + 24 a^{4} + 17 a^{3} + 15 a^{2} + 23\right)\cdot 31^{3} + \left(a^{5} + 6 a^{4} + 30 a^{3} + 17 a^{2} + 7 a + 9\right)\cdot 31^{4} + \left(18 a^{5} + 14 a^{4} + 29 a^{3} + 13 a^{2} + 20 a + 4\right)\cdot 31^{5} + \left(10 a^{5} + 15 a^{4} + 23 a^{3} + a^{2} + 5 a + 28\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 6 }$ | $=$ | \( 3 a^{5} + 23 a^{4} + 11 a^{3} + 19 a^{2} + 10 a + 12 + \left(10 a^{5} + 6 a^{4} + 28 a^{3} + 17 a^{2} + 17 a + 20\right)\cdot 31 + \left(21 a^{5} + 28 a^{4} + 24 a^{3} + 15 a^{2} + 11 a + 6\right)\cdot 31^{2} + \left(27 a^{4} + 22 a^{3} + 12 a^{2} + 4 a + 8\right)\cdot 31^{3} + \left(30 a^{4} + 27 a^{3} + 28 a^{2} + 2 a + 9\right)\cdot 31^{4} + \left(28 a^{5} + 28 a^{4} + 22 a^{3} + 26 a^{2} + 2 a + 16\right)\cdot 31^{5} + \left(13 a^{5} + 12 a^{4} + 19 a^{3} + 4 a^{2} + 14 a + 24\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 7 }$ | $=$ | \( 16 a^{5} + 28 a^{4} + 28 a^{3} + 19 a^{2} + 19 a + 2 + \left(9 a^{5} + 27 a^{4} + 25 a^{3} + 21 a^{2} + 2 a + 24\right)\cdot 31 + \left(2 a^{5} + 3 a^{4} + 18 a^{2} + 20 a + 14\right)\cdot 31^{2} + \left(20 a^{5} + 11 a^{4} + 18 a^{3} + 7 a^{2} + 19 a + 4\right)\cdot 31^{3} + \left(21 a^{5} + 5 a^{4} + 15 a^{3} + 21 a^{2} + 11 a + 24\right)\cdot 31^{4} + \left(10 a^{5} + 7 a^{4} + 24 a^{3} + 24 a^{2} + 11 a + 23\right)\cdot 31^{5} + \left(13 a^{5} + 11 a^{4} + 29 a^{3} + 8 a^{2} + 15 a + 27\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 8 }$ | $=$ | \( 4 a^{5} + 19 a^{4} + 4 a^{3} + 28 a + 1 + \left(28 a^{5} + 2 a^{4} + 27 a^{3} + 10 a^{2} + 23 a + 14\right)\cdot 31 + \left(16 a^{5} + 22 a^{3} + 24 a^{2} + 14 a + 22\right)\cdot 31^{2} + \left(9 a^{5} + 10 a^{4} + 19 a^{3} + 10 a^{2} + 21 a + 5\right)\cdot 31^{3} + \left(10 a^{5} + 24 a^{4} + 5 a^{3} + 14 a^{2} + 26 a + 20\right)\cdot 31^{4} + \left(24 a^{5} + 28 a^{4} + 4 a^{3} + 7 a^{2} + 11 a + 12\right)\cdot 31^{5} + \left(4 a^{5} + 24 a^{4} + 27 a^{3} + 17 a^{2} + 3 a + 24\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 9 }$ | $=$ | \( 11 a^{5} + 17 a^{4} + 2 a^{3} + 6 a^{2} + a + 14 + \left(12 a^{5} + 2 a^{4} + 9 a^{3} + 19 a^{2} + 18 a + 24\right)\cdot 31 + \left(9 a^{5} + 11 a^{4} + 25 a^{3} + 17 a + 21\right)\cdot 31^{2} + \left(29 a^{5} + 8 a^{4} + 9 a^{3} + 7 a^{2} + 26 a + 11\right)\cdot 31^{3} + \left(19 a^{5} + 26 a^{4} + 15 a^{3} + 3 a^{2} + 23 a + 26\right)\cdot 31^{4} + \left(2 a^{5} + a^{4} + 16 a^{3} + 29 a^{2} + 7 a + 23\right)\cdot 31^{5} + \left(18 a^{5} + 21 a^{4} + 23 a^{3} + 2 a^{2} + 5 a + 9\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 10 }$ | $=$ | \( 20 a^{4} + 20 a^{3} + 2 a^{2} + 5 a + 13 + \left(27 a^{5} + 28 a^{4} + 24 a^{3} + 2 a + 7\right)\cdot 31 + \left(15 a^{5} + 17 a^{4} + 19 a^{3} + 30 a^{2} + a + 11\right)\cdot 31^{2} + \left(6 a^{5} + 23 a^{4} + 20 a^{3} + 19 a^{2} + 15 a + 30\right)\cdot 31^{3} + \left(9 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 10 a + 19\right)\cdot 31^{4} + \left(24 a^{5} + 10 a^{4} + 30 a^{3} + 12 a^{2} + 18 a + 16\right)\cdot 31^{5} + \left(19 a^{5} + 4 a^{4} + 20 a^{3} + 11 a^{2} + a + 6\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 11 }$ | $=$ | \( 22 a^{5} + 4 a^{4} + 23 a^{3} + 3 a^{2} + 8 a + 19 + \left(3 a^{5} + 18 a^{3} + 9 a^{2} + 20 a + 22\right)\cdot 31 + \left(11 a^{5} + 26 a^{4} + 8 a^{2} + 9 a + 14\right)\cdot 31^{2} + \left(16 a^{5} + 7 a^{4} + 26 a^{2} + 13 a + 2\right)\cdot 31^{3} + \left(2 a^{5} + 23 a^{4} + 22 a^{3} + 20 a^{2} + 19 a + 19\right)\cdot 31^{4} + \left(28 a^{5} + 21 a^{4} + 24 a^{3} + 10 a^{2} + 19 a + 13\right)\cdot 31^{5} + \left(5 a^{5} + 30 a^{4} + 2 a^{3} + 25 a^{2} + 26 a + 20\right)\cdot 31^{6} +O(31^{7})\) |
$r_{ 12 }$ | $=$ | \( 6 a^{5} + 26 a^{4} + 12 a^{3} + 5 a^{2} + 20 a + 27 + \left(22 a^{5} + 28 a^{4} + 22 a^{3} + 7 a^{2} + 26 a + 26\right)\cdot 31 + \left(11 a^{5} + 18 a^{4} + 21 a^{3} + 18 a^{2} + 29 a + 19\right)\cdot 31^{2} + \left(11 a^{5} + 10 a^{4} + 8 a^{3} + 8 a^{2} + 14 a + 29\right)\cdot 31^{3} + \left(10 a^{5} + 19 a^{4} + 20 a^{3} + 13 a^{2} + 2 a + 2\right)\cdot 31^{4} + \left(8 a^{5} + 24 a^{4} + 15 a^{3} + 14 a + 28\right)\cdot 31^{5} + \left(20 a^{5} + 10 a^{4} + 10 a^{3} + 26 a^{2} + 28 a + 2\right)\cdot 31^{6} +O(31^{7})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-2$ |
$3$ | $2$ | $(1,12)(2,3)(4,5)(6,7)(8,9)(10,11)$ | $0$ |
$3$ | $2$ | $(1,6)(2,9)(3,8)(4,11)(5,10)(7,12)$ | $0$ |
$1$ | $3$ | $(1,10,8)(2,7,4)(3,6,5)(9,12,11)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,8,10)(2,4,7)(3,5,6)(9,11,12)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(3,5,6)(9,11,12)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(3,6,5)(9,12,11)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,10,8)(2,7,4)(3,5,6)(9,11,12)$ | $-1$ |
$1$ | $6$ | $(1,2,10,7,8,4)(3,11,6,9,5,12)$ | $2 \zeta_{3} + 2$ |
$1$ | $6$ | $(1,4,8,7,10,2)(3,12,5,9,6,11)$ | $-2 \zeta_{3}$ |
$2$ | $6$ | $(1,2,10,7,8,4)(3,12,5,9,6,11)$ | $1$ |
$2$ | $6$ | $(1,7)(2,8)(3,11,6,9,5,12)(4,10)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,7)(2,8)(3,12,5,9,6,11)(4,10)$ | $-\zeta_{3} - 1$ |
$3$ | $6$ | $(1,9,10,12,8,11)(2,5,7,3,4,6)$ | $0$ |
$3$ | $6$ | $(1,11,8,12,10,9)(2,6,4,3,7,5)$ | $0$ |
$3$ | $6$ | $(1,5,8,6,10,3)(2,12,4,9,7,11)$ | $0$ |
$3$ | $6$ | $(1,3,10,6,8,5)(2,11,7,9,4,12)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.