Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $1083= 3 \cdot 19^{2} $ |
| Artin number field: | Splitting field of $f= x^{12} - 2 x^{11} + 5 x^{10} - 11 x^{9} + 18 x^{8} - 43 x^{7} + 54 x^{6} - 60 x^{5} + 172 x^{4} - 261 x^{3} + 191 x^{2} - 106 x + 43 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.3_19.6t1.1c1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3 $
Roots:
| $r_{ 1 }$ | $=$ | $ 11 a^{5} + 3 a^{4} + 16 a^{3} + 2 a + 4 + \left(16 a^{5} + 5 a^{4} + 8 a^{3} + 8 a^{2} + 13 a + 12\right)\cdot 17 + \left(16 a^{5} + 16 a^{3} + 7 a^{2} + a + 3\right)\cdot 17^{2} + \left(10 a^{5} + 16 a^{4} + 12 a^{3} + 7 a^{2} + a + 10\right)\cdot 17^{3} + \left(3 a^{5} + a^{4} + 10 a^{3} + 8 a + 5\right)\cdot 17^{4} + \left(11 a^{5} + 16 a^{4} + 5 a^{2} + 9 a + 15\right)\cdot 17^{5} + \left(14 a^{4} + 5 a^{3} + 8 a^{2} + 13 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 3 a^{5} + 2 a^{4} + a^{3} + 7 a^{2} + 12 a + 6 + \left(10 a^{5} + 15 a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 5\right)\cdot 17 + \left(4 a^{5} + 14 a^{4} + 16 a^{3} + 10 a^{2} + 11 a + 7\right)\cdot 17^{2} + \left(13 a^{5} + 6 a^{4} + 10 a^{3} + 15 a^{2} + 2 a + 9\right)\cdot 17^{3} + \left(a^{5} + 10 a^{4} + 3 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 17^{4} + \left(14 a^{5} + 15 a^{4} + 5 a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 17^{5} + \left(4 a^{5} + 4 a^{3} + 14 a^{2} + 5 a + 3\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 6 a^{5} + 3 a^{4} + 4 a^{3} + 4 a^{2} + 10 a + 14 + \left(4 a^{5} + 8 a^{4} + 10 a^{3} + 14 a^{2} + 12 a + 15\right)\cdot 17 + \left(7 a^{5} + 14 a^{3} + 6 a^{2} + 14 a + 13\right)\cdot 17^{2} + \left(10 a^{5} + 13 a^{4} + 4 a^{3} + 16 a^{2} + 16 a + 15\right)\cdot 17^{3} + \left(3 a^{5} + 5 a^{4} + 3 a^{3} + 16 a^{2} + 3 a + 2\right)\cdot 17^{4} + \left(12 a^{5} + 3 a^{4} + 5 a^{3} + 15 a^{2} + 5 a + 5\right)\cdot 17^{5} + \left(7 a^{5} + 7 a^{4} + 13 a^{3} + 14 a^{2} + 7 a + 5\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 6 a^{4} + 6 a^{3} + 13 a^{2} + 11 a + 4 + \left(16 a^{5} + 12 a^{4} + 5 a^{3} + 14 a^{2} + 5\right)\cdot 17 + \left(14 a^{5} + 11 a^{4} + a^{3} + 16 a^{2} + 10 a + 6\right)\cdot 17^{2} + \left(16 a^{5} + 11 a^{4} + 13 a^{3} + 2 a^{2} + 11 a + 7\right)\cdot 17^{3} + \left(7 a^{5} + 16 a^{4} + 3 a^{3} + 3 a^{2} + 12 a + 3\right)\cdot 17^{4} + \left(7 a^{5} + 15 a^{4} + 10 a^{3} + 16 a^{2} + 4\right)\cdot 17^{5} + \left(14 a^{5} + 6 a^{4} + 10 a^{2} + 13 a + 15\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 5 a^{5} + 12 a^{4} + 10 a^{3} + 11 a^{2} + 3 a + 16 + \left(a^{5} + 16 a^{4} + 16 a^{3} + 10 a^{2} + a + 5\right)\cdot 17 + \left(16 a^{5} + 9 a^{4} + a^{3} + 8 a^{2} + 14 a + 11\right)\cdot 17^{2} + \left(3 a^{5} + 2 a^{4} + 13 a^{3} + a^{2} + 9 a + 7\right)\cdot 17^{3} + \left(15 a^{5} + 8 a^{4} + 11 a^{2} + 16 a + 12\right)\cdot 17^{4} + \left(12 a^{5} + 11 a^{4} + a^{3} + 14 a^{2} + 12 a + 9\right)\cdot 17^{5} + \left(9 a^{5} + 11 a^{4} + 10 a^{3} + 12 a^{2} + 15 a + 4\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 9 a^{5} + 3 a^{4} + 3 a^{3} + 4 a + 16 + \left(14 a^{5} + 10 a^{4} + 16 a^{3} + 9 a^{2} + 6 a + 5\right)\cdot 17 + \left(2 a^{5} + 4 a^{4} + 12 a^{3} + 5 a^{2} + 4 a + 1\right)\cdot 17^{2} + \left(7 a^{5} + 7 a^{4} + 10 a^{3} + 16 a^{2} + 16 a + 8\right)\cdot 17^{3} + \left(11 a^{5} + 9 a^{4} + 6 a^{3} + 11 a^{2} + 4 a + 7\right)\cdot 17^{4} + \left(7 a^{5} + 5 a^{4} + 12 a^{3} + 10 a^{2} + 5 a + 13\right)\cdot 17^{5} + \left(12 a^{5} + 12 a^{4} + 16 a^{3} + 13 a^{2} + a + 6\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ a^{5} + 6 a^{4} + 14 a^{3} + 4 a^{2} + 10 a + 9 + \left(9 a^{4} + a^{3} + 12 a^{2} + 4 a + 1\right)\cdot 17 + \left(15 a^{5} + a^{4} + 14 a^{3} + 15 a^{2} + 13 a + 8\right)\cdot 17^{2} + \left(4 a^{5} + 4 a^{4} + 6 a^{3} + 5 a^{2} + 4 a + 12\right)\cdot 17^{3} + \left(15 a^{5} + 12 a^{4} + 13 a^{3} + 1\right)\cdot 17^{4} + \left(11 a^{5} + 10 a^{4} + 14 a^{3} + 12 a^{2} + 15 a + 7\right)\cdot 17^{5} + \left(9 a^{5} + 14 a^{4} + 15 a^{3} + 7 a^{2} + 14 a + 5\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 15 a^{5} + 2 a^{4} + 9 a^{3} + 14 a^{2} + 2 a + 1 + \left(7 a^{5} + 3 a^{4} + 15 a^{3} + 7 a^{2} + 16\right)\cdot 17 + \left(13 a^{5} + 3 a^{4} + 10 a^{3} + 11 a + 16\right)\cdot 17^{2} + \left(a^{5} + 9 a^{4} + 9 a^{3} + a^{2} + 9 a + 5\right)\cdot 17^{3} + \left(16 a^{5} + 16 a^{4} + 13 a^{3} + 3 a^{2} + 16 a + 14\right)\cdot 17^{4} + \left(3 a^{5} + 8 a^{3} + 11 a^{2} + 3 a + 16\right)\cdot 17^{5} + \left(2 a^{5} + 6 a^{4} + 15 a^{3} + 13 a + 9\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 10 a^{5} + 8 a^{4} + a^{3} + 5 a^{2} + 2 a + 6 + \left(14 a^{5} + 14 a^{4} + 10 a^{3} + 5 a^{2} + 14 a + 12\right)\cdot 17 + \left(2 a^{5} + 8 a^{4} + 3 a^{3} + 4 a^{2} + 12 a + 14\right)\cdot 17^{2} + \left(4 a^{5} + 6 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 17^{3} + \left(a^{5} + 11 a^{4} + 4 a^{3} + 16 a^{2} + 16 a + 6\right)\cdot 17^{4} + \left(6 a^{5} + 4 a^{4} + a^{3} + 12 a^{2} + 4 a + 3\right)\cdot 17^{5} + \left(5 a^{5} + 8 a^{4} + 2 a^{3} + 10 a^{2} + 3 a + 5\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 8 a^{4} + 8 a^{3} + 15 a^{2} + 11 a + 16 + \left(4 a^{5} + 7 a^{4} + 15 a^{3} + 15 a^{2} + 5 a\right)\cdot 17 + \left(11 a^{5} + 8 a^{4} + 4 a^{3} + a^{2} + 16 a + 12\right)\cdot 17^{2} + \left(16 a^{4} + 3 a^{3} + 8 a^{2} + 6 a + 15\right)\cdot 17^{3} + \left(9 a^{5} + 2 a^{4} + 3 a^{3} + 12 a^{2} + 6 a + 9\right)\cdot 17^{4} + \left(2 a^{5} + 14 a^{4} + 13 a^{3} + 12 a^{2} + 15 a + 2\right)\cdot 17^{5} + \left(11 a^{5} + 13 a^{4} + 14 a^{3} + 5 a\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 16 a^{5} + 9 a^{4} + a^{3} + 15 a^{2} + 5 a + 16 + \left(14 a^{5} + a^{4} + 2 a^{3} + a^{2} + 8 a + 9\right)\cdot 17 + \left(a^{5} + 4 a^{4} + 8 a^{3} + 8 a^{2} + 16 a + 3\right)\cdot 17^{2} + \left(16 a^{5} + 12 a^{4} + 14 a^{3} + 2 a^{2} + 5 a + 5\right)\cdot 17^{3} + \left(15 a^{5} + a^{4} + 8 a^{3} + 16 a^{2}\right)\cdot 17^{4} + \left(13 a^{5} + 13 a^{4} + 3 a^{3} + 4 a^{2} + 4 a + 6\right)\cdot 17^{5} + \left(6 a^{5} + 2 a^{4} + 7 a^{2} + 5 a + 8\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 9 a^{5} + 6 a^{4} + 12 a^{3} + 14 a^{2} + 13 a + 13 + \left(14 a^{5} + 15 a^{4} + 10 a^{3} + 8 a^{2} + 11 a + 10\right)\cdot 17 + \left(11 a^{5} + 16 a^{4} + 13 a^{3} + 15 a^{2} + 9 a + 2\right)\cdot 17^{2} + \left(11 a^{5} + 12 a^{4} + 5 a^{3} + a^{2} + 13 a + 9\right)\cdot 17^{3} + \left(4 a^{4} + 12 a^{3} + 7 a^{2} + 6 a + 3\right)\cdot 17^{4} + \left(15 a^{5} + 7 a^{4} + 8 a^{3} + 16 a^{2} + 2 a + 5\right)\cdot 17^{5} + \left(16 a^{5} + 2 a^{4} + 3 a^{3} + 16 a^{2} + 3 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,9)(2,6)(3,7)(4,5)(8,12)(10,11)$ | $-2$ |
| $3$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,12)(4,11)(5,10)(6,8)(7,9)$ | $0$ |
| $1$ | $3$ | $(1,2,5)(3,12,10)(4,9,6)(7,8,11)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,2)(3,10,12)(4,6,9)(7,11,8)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(3,10,12)(7,11,8)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(3,12,10)(7,8,11)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,2)(3,12,10)(4,6,9)(7,8,11)$ | $-1$ |
| $1$ | $6$ | $(1,6,5,9,2,4)(3,8,10,7,12,11)$ | $2 \zeta_{3} + 2$ |
| $1$ | $6$ | $(1,4,2,9,5,6)(3,11,12,7,10,8)$ | $-2 \zeta_{3}$ |
| $2$ | $6$ | $(1,9)(2,6)(3,11,12,7,10,8)(4,5)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,9)(2,6)(3,8,10,7,12,11)(4,5)$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,4,2,9,5,6)(3,8,10,7,12,11)$ | $1$ |
| $3$ | $6$ | $(1,8,5,7,2,11)(3,6,10,9,12,4)$ | $0$ |
| $3$ | $6$ | $(1,11,2,7,5,8)(3,4,12,9,10,6)$ | $0$ |
| $3$ | $6$ | $(1,12,5,3,2,10)(4,7,6,11,9,8)$ | $0$ |
| $3$ | $6$ | $(1,10,2,3,5,12)(4,8,9,11,6,7)$ | $0$ |