Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_4$ |
| Conductor: | $1352= 2^{3} \cdot 13^{2} $ |
| Artin number field: | Splitting field of $f= x^{12} - 4 x^{10} - 14 x^{9} + 6 x^{8} + 42 x^{7} + 11 x^{6} - 42 x^{5} - 29 x^{4} + 148 x^{3} + 15 x^{2} - 134 x + 107 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $S_3 \times C_4$ |
| Parity: | Odd |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{4} + 2 x^{2} + 11 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 3 a^{3} + 10 a^{2} + 10 a + 15 + \left(14 a^{3} + 9 a^{2} + 11 a + 10\right)\cdot 19 + \left(9 a^{3} + 7 a + 1\right)\cdot 19^{2} + \left(8 a^{3} + 7 a^{2} + 6 a + 3\right)\cdot 19^{3} + \left(5 a^{3} + 18 a + 7\right)\cdot 19^{4} + \left(6 a^{3} + 7 a + 8\right)\cdot 19^{5} + \left(5 a^{3} + 15 a^{2} + 14 a + 9\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 16 a^{3} + 12 a + 11 + \left(7 a^{3} + 4 a^{2} + 7 a + 11\right)\cdot 19 + \left(16 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 19^{2} + \left(12 a^{2} + a + 12\right)\cdot 19^{3} + \left(10 a^{3} + 3 a^{2} + 7 a\right)\cdot 19^{4} + \left(18 a^{3} + 9 a^{2} + 11 a + 2\right)\cdot 19^{5} + \left(4 a^{3} + 11 a^{2} + 18 a + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 4 a^{3} + 18 a^{2} + 10 a + 17 + \left(15 a^{3} + 12 a^{2} + 9 a + 3\right)\cdot 19 + \left(2 a^{3} + 5 a^{2} + 12 a + 11\right)\cdot 19^{2} + \left(6 a^{3} + 17 a^{2} + 13 a + 17\right)\cdot 19^{3} + \left(16 a^{3} + 12 a^{2} + 10 a + 4\right)\cdot 19^{4} + \left(4 a^{2} + a + 6\right)\cdot 19^{5} + \left(2 a^{3} + 5 a^{2} + 15 a + 15\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ a^{3} + 2 a^{2} + 11 a + 18 + \left(8 a^{3} + 9 a^{2} + 10 a + 5\right)\cdot 19 + \left(11 a^{3} + 6 a^{2} + 15 a + 6\right)\cdot 19^{2} + \left(14 a^{3} + 11 a^{2} + 11 a + 18\right)\cdot 19^{3} + \left(12 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 19^{4} + \left(11 a^{3} + 14 a^{2} + 12 a + 5\right)\cdot 19^{5} + \left(4 a^{3} + 3 a^{2} + 7 a + 12\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 8 a^{3} + 18 a + 12 + \left(11 a^{3} + 7 a^{2} + 11 a + 12\right)\cdot 19 + \left(5 a^{3} + 2 a^{2} + 1\right)\cdot 19^{2} + \left(2 a^{3} + 16 a^{2} + 5 a + 7\right)\cdot 19^{3} + \left(7 a^{3} + 14 a^{2} + 18 a + 12\right)\cdot 19^{4} + \left(6 a^{3} + a^{2} + 7 a + 15\right)\cdot 19^{5} + \left(17 a^{3} + 17 a^{2} + 11 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 14 a^{3} + 8 a + 15 + \left(18 a^{3} + 8 a^{2} + 18 a + 13\right)\cdot 19 + \left(15 a^{3} + 6 a^{2} + 15 a + 6\right)\cdot 19^{2} + \left(15 a^{3} + 9 a^{2} + 12 a + 18\right)\cdot 19^{3} + \left(a^{3} + 12 a + 5\right)\cdot 19^{4} + \left(13 a^{3} + 8 a^{2} + 18 a + 1\right)\cdot 19^{5} + \left(15 a^{3} + 9 a^{2} + 7 a + 18\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 7 a^{3} + 16 a^{2} + 13 a + 5 + \left(4 a^{3} + 6 a^{2} + 5 a + 14\right)\cdot 19 + \left(a^{2} + 7 a + 10\right)\cdot 19^{2} + \left(6 a^{3} + 12 a^{2} + 4 a + 16\right)\cdot 19^{3} + \left(7 a^{3} + 3 a + 17\right)\cdot 19^{4} + \left(4 a^{3} + 7 a^{2} + 10 a + 1\right)\cdot 19^{5} + \left(17 a^{3} + 2 a^{2} + 12 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 14 a^{3} + 12 a^{2} + 15 a + 7 + \left(5 a^{3} + 4 a^{2} + 14 a + 1\right)\cdot 19 + \left(9 a^{3} + 11 a^{2} + 9 a + 3\right)\cdot 19^{2} + \left(5 a^{3} + 2 a^{2} + 9 a + 15\right)\cdot 19^{3} + \left(16 a^{3} + 13 a^{2}\right)\cdot 19^{4} + \left(15 a^{3} + 5 a^{2} + 18 a + 17\right)\cdot 19^{5} + \left(5 a^{3} + 8 a^{2} + 5 a + 7\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 14 a^{3} + 18 a^{2} + 17 a + 3 + \left(14 a^{3} + 15 a^{2} + 17 a + 9\right)\cdot 19 + \left(4 a^{3} + 6 a^{2} + 9 a + 1\right)\cdot 19^{2} + \left(17 a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 19^{3} + \left(8 a^{3} + 5 a + 7\right)\cdot 19^{4} + \left(6 a^{3} + 5 a + 7\right)\cdot 19^{5} + \left(12 a^{3} + 10 a^{2} + 15 a + 10\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ a^{3} + 4 a^{2} + 15 a + 10 + \left(11 a^{3} + 11 a^{2} + 6 a + 2\right)\cdot 19 + \left(16 a^{3} + 18 a + 7\right)\cdot 19^{2} + \left(13 a^{3} + 4 a^{2} + 16\right)\cdot 19^{3} + \left(11 a^{3} + 14 a^{2} + 3 a + 10\right)\cdot 19^{4} + \left(8 a^{3} + 2 a^{2} + 11 a + 13\right)\cdot 19^{5} + \left(3 a^{3} + 14 a^{2} + 10 a + 7\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 15 a^{3} + 5 a^{2} + 13 a + 13 + \left(12 a^{3} + 17 a^{2} + 5\right)\cdot 19 + \left(11 a^{3} + 17 a^{2} + 12 a + 10\right)\cdot 19^{2} + \left(15 a^{3} + 7 a^{2} + 11 a + 18\right)\cdot 19^{3} + \left(a^{3} + 4 a^{2} + 16 a\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 18 a + 16\right)\cdot 19^{5} + \left(10 a^{3} + 8 a^{2} + 12 a + 1\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 17 a^{3} + 10 a^{2} + 10 a + 7 + \left(8 a^{3} + 7 a^{2} + 17 a + 3\right)\cdot 19 + \left(9 a^{3} + 6 a^{2} + a + 5\right)\cdot 19^{2} + \left(7 a^{3} + 4 a^{2} + 5 a + 6\right)\cdot 19^{3} + \left(14 a^{3} + 5 a^{2} + 15 a\right)\cdot 19^{4} + \left(17 a^{3} + 6 a^{2} + 9 a\right)\cdot 19^{5} + \left(14 a^{3} + 8 a^{2} + 6\right)\cdot 19^{6} +O\left(19^{ 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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,11)(5,8)(6,12)(9,10)$ | $-2$ | $-2$ |
| $3$ | $2$ | $(1,9)(2,8)(3,10)(4,11)(5,7)(6,12)$ | $0$ | $0$ |
| $3$ | $2$ | $(1,10)(2,5)(3,9)(7,8)$ | $0$ | $0$ |
| $2$ | $3$ | $(1,10,11)(2,6,5)(3,9,4)(7,12,8)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,8,3,5)(2,10,7,9)(4,6,11,12)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,3,8)(2,9,7,10)(4,12,11,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $3$ | $4$ | $(1,12,3,6)(2,10,7,9)(4,5,11,8)$ | $0$ | $0$ |
| $3$ | $4$ | $(1,6,3,12)(2,9,7,10)(4,8,11,5)$ | $0$ | $0$ |
| $2$ | $6$ | $(1,4,10,3,11,9)(2,8,6,7,5,12)$ | $1$ | $1$ |
| $2$ | $12$ | $(1,2,4,8,10,6,3,7,11,5,9,12)$ | $-\zeta_{4}$ | $\zeta_{4}$ |
| $2$ | $12$ | $(1,7,4,5,10,12,3,2,11,8,9,6)$ | $\zeta_{4}$ | $-\zeta_{4}$ |