Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $1352= 2^{3} \cdot 13^{2} $ |
| Artin number field: | Splitting field of $f= x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 61 x^{8} - 30 x^{7} + 202 x^{6} - 74 x^{5} + 347 x^{4} + 36 x^{3} + 130 x^{2} + 132 x + 27 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.2e3_13.6t1.4c2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 4 a^{5} + 16 a^{4} + 8 a^{3} + 8 a^{2} + 7 a + 14 + \left(7 a^{5} + 5 a^{4} + 5 a^{3} + 4 a^{2} + a + 12\right)\cdot 19 + \left(6 a^{5} + 12 a^{4} + 2 a^{3} + 18 a^{2} + 14\right)\cdot 19^{2} + \left(8 a^{5} + 18 a^{4} + 3 a^{3} + 10 a^{2} + 13 a + 4\right)\cdot 19^{3} + \left(7 a^{4} + 12 a^{2} + 11 a + 12\right)\cdot 19^{4} + \left(2 a^{5} + 4 a^{4} + 10 a^{3} + 2 a^{2} + 18 a\right)\cdot 19^{5} + \left(12 a^{5} + 10 a^{4} + 2 a^{3} + 10 a^{2} + 5\right)\cdot 19^{6} + \left(17 a^{5} + 3 a^{4} + 3 a^{3} + 9 a^{2} + 16 a + 1\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 9 a^{5} + 8 a^{4} + 11 a^{3} + a^{2} + 8 a + 13 + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + 10 a + 18\right)\cdot 19 + \left(11 a^{5} + 10 a^{4} + 4 a^{3} + 11 a^{2} + 15 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 18 a^{4} + 14 a^{3} + 4 a^{2} + 17 a + 14\right)\cdot 19^{3} + \left(2 a^{5} + 16 a^{4} + 16 a^{3} + 6 a^{2} + 14 a + 9\right)\cdot 19^{4} + \left(15 a^{5} + 10 a^{4} + a^{3} + 8 a^{2} + 14 a + 14\right)\cdot 19^{5} + \left(9 a^{5} + 3 a^{4} + 17 a^{3} + 17 a^{2} + 8 a + 9\right)\cdot 19^{6} + \left(2 a^{5} + 4 a^{4} + 11 a^{3} + 6 a^{2} + 18 a + 4\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 16 a^{5} + 13 a^{4} + 7 a^{3} + 6 a^{2} + 17 a + 1 + \left(14 a^{5} + 7 a^{4} + 16 a^{3} + 11 a^{2} + 16 a + 18\right)\cdot 19 + \left(13 a^{5} + 4 a^{4} + 8 a^{3} + 13 a^{2} + 4 a + 2\right)\cdot 19^{2} + \left(13 a^{5} + 7 a^{4} + a^{3} + 17 a^{2} + a + 7\right)\cdot 19^{3} + \left(14 a^{5} + 14 a^{4} + 15 a^{3} + 9 a + 6\right)\cdot 19^{4} + \left(4 a^{5} + 12 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 19^{5} + \left(11 a^{5} + 2 a^{4} + 5 a^{3} + 12 a^{2} + 6 a + 1\right)\cdot 19^{6} + \left(a^{5} + 3 a^{4} + 14 a^{3} + 17 a^{2} + 8 a + 5\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 8 a^{5} + 9 a^{4} + 15 a^{3} + 9 a^{2} + 6 a + 11 + \left(16 a^{5} + 3 a^{4} + 2 a^{3} + a^{2} + 5 a + 9\right)\cdot 19 + \left(4 a^{5} + a^{4} + 16 a^{3} + 16 a^{2} + 5 a + 11\right)\cdot 19^{2} + \left(16 a^{5} + 16 a^{4} + 10 a^{2} + 11 a + 11\right)\cdot 19^{3} + \left(8 a^{5} + 12 a^{4} + 2 a^{3} + 18 a^{2} + 2 a + 14\right)\cdot 19^{4} + \left(6 a^{5} + 15 a^{4} + 17 a^{3} + 12 a^{2} + 14 a + 1\right)\cdot 19^{5} + \left(12 a^{5} + 6 a^{4} + 2 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 19^{6} + \left(a^{5} + 5 a^{4} + 13 a^{3} + 9 a^{2} + 15 a + 1\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 5 }$ | $=$ | $ a^{4} + 11 a^{3} + 12 a^{2} + 15 a + 11 + \left(5 a^{5} + 18 a^{4} + 5 a^{3} + 7 a^{2} + 13 a + 11\right)\cdot 19 + \left(8 a^{5} + 12 a^{4} + 7 a^{3} + 17 a^{2} + 4 a + 10\right)\cdot 19^{2} + \left(17 a^{5} + 10 a^{4} + 12 a^{3} + 2 a^{2} + 3 a + 16\right)\cdot 19^{3} + \left(18 a^{5} + 16 a^{4} + 3 a^{3} + 14 a^{2} + 2 a + 9\right)\cdot 19^{4} + \left(14 a^{5} + 4 a^{4} + 16 a^{3} + 9 a^{2} + 6 a + 15\right)\cdot 19^{5} + \left(7 a^{5} + 7 a^{4} + 2 a^{3} + 2 a^{2} + 13 a + 15\right)\cdot 19^{6} + \left(7 a^{5} + 13 a^{4} + 13 a^{3} + 14 a^{2} + 9 a + 13\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 17 a^{5} + 16 a^{4} + 10 a^{3} + 10 a^{2} + 7 a + 18 + \left(14 a^{5} + 18 a^{4} + 7 a^{2} + 8 a + 6\right)\cdot 19 + \left(18 a^{5} + 18 a^{4} + 18 a^{3} + 8 a^{2} + 9 a + 8\right)\cdot 19^{2} + \left(9 a^{5} + 2 a^{4} + 18 a^{3} + 15 a^{2} + 4 a\right)\cdot 19^{3} + \left(8 a^{5} + a^{4} + 15 a^{3} + a^{2} + 10 a + 17\right)\cdot 19^{4} + \left(13 a^{5} + 3 a^{4} + 4 a^{3} + 12 a^{2} + a + 5\right)\cdot 19^{5} + \left(17 a^{5} + 10 a^{4} + 12 a^{3} + 9 a^{2} + 17 a + 16\right)\cdot 19^{6} + \left(9 a^{5} + 13 a^{4} + 7 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 7 }$ | $=$ | $ a^{5} + 4 a^{4} + 2 a^{3} + 7 a^{2} + a + 2 + \left(a^{5} + 11 a^{3} + 4 a^{2} + 9 a + 17\right)\cdot 19 + \left(14 a^{5} + 3 a^{4} + 13 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 14 a^{4} + 16 a^{3} + 17 a + 13\right)\cdot 19^{3} + \left(9 a^{5} + 18 a^{4} + a^{3} + 9 a^{2} + 6 a + 18\right)\cdot 19^{4} + \left(7 a^{5} + 14 a^{4} + 16 a^{3} + 11 a^{2} + 4 a + 18\right)\cdot 19^{5} + \left(14 a^{5} + 13 a^{3} + 14 a^{2} + 11 a + 11\right)\cdot 19^{6} + \left(11 a^{5} + 16 a^{4} + 16 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 7 a^{5} + 10 a^{4} + 11 a^{3} + 18 a^{2} + 9 a + 13 + \left(10 a^{5} + 16 a^{3} + 4 a^{2} + 6 a + 17\right)\cdot 19 + \left(7 a^{5} + 10 a^{4} + a^{2} + 5 a + 11\right)\cdot 19^{2} + \left(10 a^{5} + 18 a^{4} + 7 a^{3} + 8 a^{2} + 14 a + 7\right)\cdot 19^{3} + \left(15 a^{5} + 8 a^{4} + 7 a^{3} + 18 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(15 a^{5} + 16 a^{4} + 18 a^{3} + 6 a^{2} + 15 a + 12\right)\cdot 19^{5} + \left(10 a^{5} + 3 a^{4} + 3 a^{3} + 13 a^{2} + 14 a + 1\right)\cdot 19^{6} + \left(14 a^{5} + a^{4} + 5 a^{3} + 18 a^{2} + 8 a + 12\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 10 a^{5} + 9 a^{4} + 9 a^{3} + 17 a^{2} + 2 a + 6 + \left(13 a^{5} + 8 a^{4} + 6 a^{3} + 2 a^{2} + 4 a + 12\right)\cdot 19 + \left(16 a^{5} + 7 a^{4} + 15 a^{3} + 17 a^{2} + 8 a + 2\right)\cdot 19^{2} + \left(7 a^{5} + 18 a^{4} + 4 a^{3} + 14 a^{2} + 13 a + 6\right)\cdot 19^{3} + \left(15 a^{5} + 13 a^{4} + 16 a^{3} + 11 a^{2} + 7 a + 18\right)\cdot 19^{4} + \left(12 a^{5} + 18 a^{4} + 18 a^{3} + 8 a^{2} + 13 a + 4\right)\cdot 19^{5} + \left(4 a^{5} + 13 a^{4} + 14 a^{3} + a^{2} + 8\right)\cdot 19^{6} + \left(7 a^{5} + 18 a^{4} + 5 a^{3} + 6 a^{2} + 12 a + 14\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 11 a^{5} + a^{3} + 7 a^{2} + 7 a + 14 + \left(13 a^{5} + 13 a^{4} + 13 a^{2} + 17 a + 16\right)\cdot 19 + \left(a^{5} + 13 a^{4} + 17 a^{3} + 11 a^{2} + 12 a + 14\right)\cdot 19^{2} + \left(2 a^{5} + 18 a^{4} + 9 a^{3} + 7 a^{2} + 16 a + 9\right)\cdot 19^{3} + \left(2 a^{5} + a^{4} + 17 a^{3} + 7 a^{2} + 5 a + 11\right)\cdot 19^{4} + \left(5 a^{5} + 5 a^{4} + 14 a^{3} + 13 a^{2} + 16 a + 8\right)\cdot 19^{5} + \left(6 a^{5} + 11 a^{4} + 2 a^{3} + 6 a^{2} + 8 a + 2\right)\cdot 19^{6} + \left(13 a^{5} + 18 a^{4} + 18 a^{3} + 11 a^{2} + 13 a + 15\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 14 a^{5} + 8 a^{4} + a^{3} + 14 a^{2} + 2 a + 12 + \left(13 a^{5} + 16 a^{4} + 13 a^{3} + 6 a^{2} + 10 a + 11\right)\cdot 19 + \left(2 a^{5} + 13 a^{4} + 3 a^{2} + 6 a + 3\right)\cdot 19^{2} + \left(8 a^{5} + 16 a^{4} + 14 a^{3} + 5 a^{2} + 14 a + 5\right)\cdot 19^{3} + \left(17 a^{5} + 17 a^{4} + 12 a^{3} + 14 a^{2} + 8 a + 12\right)\cdot 19^{4} + \left(a^{5} + 11 a^{4} + a^{3} + 6 a^{2} + 16 a + 17\right)\cdot 19^{5} + \left(4 a^{5} + a^{4} + 13 a^{3} + 14 a^{2} + 13 a + 7\right)\cdot 19^{6} + \left(5 a^{5} + 11 a^{3} + 17 a^{2} + 13 a + 4\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 17 a^{5} + a^{4} + 9 a^{3} + 5 a^{2} + 14 a + 1 + \left(2 a^{5} + 15 a^{4} + 9 a^{3} + a^{2} + 10 a + 18\right)\cdot 19 + \left(8 a^{5} + 5 a^{4} + 9 a^{3} + 5 a^{2} + 13 a + 14\right)\cdot 19^{2} + \left(10 a^{5} + 10 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 16\right)\cdot 19^{3} + \left(a^{4} + 4 a^{3} + 17 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(14 a^{5} + 14 a^{4} + 8 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 19^{5} + \left(2 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 19^{6} + \left(2 a^{5} + 16 a^{4} + 12 a^{3} + 4 a^{2} + a + 4\right)\cdot 19^{7} +O\left(19^{ 8 }\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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,11)(3,10)(4,9)(5,8)(7,12)$ | $-2$ |
| $3$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $0$ |
| $3$ | $2$ | $(1,12)(2,5)(3,4)(6,7)(8,11)(9,10)$ | $0$ |
| $1$ | $3$ | $(1,10,5)(2,12,9)(3,8,6)(4,11,7)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,10)(2,9,12)(3,6,8)(4,7,11)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,10,5)(3,8,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,10)(3,6,8)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,10,5)(2,9,12)(3,8,6)(4,7,11)$ | $-1$ |
| $1$ | $6$ | $(1,8,10,6,5,3)(2,4,12,11,9,7)$ | $2 \zeta_{3} + 2$ |
| $1$ | $6$ | $(1,3,5,6,10,8)(2,7,9,11,12,4)$ | $-2 \zeta_{3}$ |
| $2$ | $6$ | $(1,3,5,6,10,8)(2,11)(4,9)(7,12)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,8,10,6,5,3)(2,11)(4,9)(7,12)$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,3,5,6,10,8)(2,4,12,11,9,7)$ | $1$ |
| $3$ | $6$ | $(1,11,10,7,5,4)(2,3,12,8,9,6)$ | $0$ |
| $3$ | $6$ | $(1,4,5,7,10,11)(2,6,9,8,12,3)$ | $0$ |
| $3$ | $6$ | $(1,2,10,12,5,9)(3,7,8,4,6,11)$ | $0$ |
| $3$ | $6$ | $(1,9,5,12,10,2)(3,11,6,4,8,7)$ | $0$ |