Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_4$ |
| Conductor: | $3328= 2^{8} \cdot 13 $ |
| Artin number field: | Splitting field of $f= x^{12} - 4 x^{10} + 6 x^{8} + 8 x^{6} - 7 x^{4} - 4 x^{2} + 2 $ 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 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{4} + 2 x^{2} + 11 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 6 a^{2} + 10 a + 6 + \left(11 a^{3} + 8 a^{2} + 8 a + 18\right)\cdot 19 + \left(9 a^{3} + 15 a^{2} + 10 a + 8\right)\cdot 19^{2} + \left(10 a^{3} + 9 a^{2} + 12 a + 1\right)\cdot 19^{3} + \left(a^{3} + a^{2} + 5 a\right)\cdot 19^{4} + \left(12 a^{3} + 8 a^{2} + 16 a + 8\right)\cdot 19^{5} + \left(a^{3} + 12 a + 4\right)\cdot 19^{6} + \left(15 a^{3} + 7 a^{2} + 18 a + 3\right)\cdot 19^{7} + \left(4 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 14 a^{3} + 13 a^{2} + 6 a + 5 + \left(a^{3} + 2 a^{2} + 5 a + 12\right)\cdot 19 + \left(3 a^{3} + 6 a^{2} + 10 a + 12\right)\cdot 19^{2} + \left(8 a^{3} + 4 a^{2} + 13 a + 14\right)\cdot 19^{3} + \left(9 a^{3} + 17 a^{2} + 17 a + 4\right)\cdot 19^{4} + \left(a^{3} + 14 a^{2} + 11 a + 8\right)\cdot 19^{5} + \left(6 a^{3} + 5 a^{2} + 7 a + 8\right)\cdot 19^{6} + \left(4 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 19^{7} + \left(5 a^{3} + 12 a\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 12 a^{3} + 7 a^{2} + 10 a + 11 + \left(11 a^{3} + 18 a + 15\right)\cdot 19 + \left(12 a^{3} + 18 a^{2} + 11 a + 12\right)\cdot 19^{2} + \left(12 a^{3} + 17 a^{2} + 2 a + 3\right)\cdot 19^{3} + \left(7 a^{3} + 7 a^{2} + 4 a + 14\right)\cdot 19^{4} + \left(10 a^{3} + a^{2} + 4 a + 1\right)\cdot 19^{5} + \left(13 a^{3} + 12 a^{2} + 18 a + 5\right)\cdot 19^{6} + \left(12 a^{3} + 5 a^{2} + 18 a + 1\right)\cdot 19^{7} + \left(15 a^{3} + 18 a^{2} + 14 a + 5\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 4 }$ | $=$ | $ a^{3} + 2 a^{2} + 9 a + 15 + \left(13 a^{3} + 17 a^{2} + 4 a + 10\right)\cdot 19 + \left(4 a^{3} + 12 a^{2} + 13\right)\cdot 19^{2} + \left(17 a^{3} + 16 a^{2} + 6 a + 11\right)\cdot 19^{3} + \left(15 a^{3} + 2 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(16 a^{3} + 9 a^{2} + 5\right)\cdot 19^{5} + \left(6 a^{3} + 4 a^{2} + 13 a + 4\right)\cdot 19^{6} + \left(4 a^{3} + 13 a^{2} + 7 a + 11\right)\cdot 19^{7} + \left(8 a^{3} + 13 a^{2} + 5\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 7 a^{3} + 8 a + 15 + \left(18 a^{3} + 9 a^{2} + 17 a + 3\right)\cdot 19 + \left(9 a^{3} + 14 a^{2} + 6 a + 11\right)\cdot 19^{2} + \left(14 a^{3} + 9 a^{2} + 3 a + 15\right)\cdot 19^{3} + \left(15 a^{3} + 18 a^{2} + 14 a + 10\right)\cdot 19^{4} + \left(a^{3} + 2 a^{2} + 8\right)\cdot 19^{5} + \left(10 a^{3} + 11 a^{2} + 18\right)\cdot 19^{6} + \left(a^{3} + 7 a^{2} + 8 a + 5\right)\cdot 19^{7} + \left(7 a^{3} + 15 a^{2} + 16 a + 7\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 15 a^{3} + 10 a^{2} + 13 a + 15 + \left(3 a^{3} + 11 a^{2} + a + 4\right)\cdot 19 + \left(6 a^{3} + 11 a^{2} + 9 a + 15\right)\cdot 19^{2} + \left(4 a^{3} + 6 a^{2} + 6 a + 13\right)\cdot 19^{3} + \left(3 a^{3} + 7 a + 12\right)\cdot 19^{4} + \left(4 a^{3} + 2 a^{2} + 14 a + 12\right)\cdot 19^{5} + \left(15 a^{3} + 18 a^{2} + 4 a + 10\right)\cdot 19^{6} + \left(8 a^{3} + 2 a^{2} + 17 a + 18\right)\cdot 19^{7} + \left(15 a^{3} + 12 a^{2} + 13 a + 1\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 13 a^{2} + 9 a + 13 + \left(8 a^{3} + 10 a^{2} + 10 a\right)\cdot 19 + \left(9 a^{3} + 3 a^{2} + 8 a + 10\right)\cdot 19^{2} + \left(8 a^{3} + 9 a^{2} + 6 a + 17\right)\cdot 19^{3} + \left(17 a^{3} + 17 a^{2} + 13 a + 18\right)\cdot 19^{4} + \left(6 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 19^{5} + \left(17 a^{3} + 18 a^{2} + 6 a + 14\right)\cdot 19^{6} + \left(3 a^{3} + 11 a^{2} + 15\right)\cdot 19^{7} + \left(14 a^{3} + 9 a^{2} + 12 a + 12\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 5 a^{3} + 6 a^{2} + 13 a + 14 + \left(17 a^{3} + 16 a^{2} + 13 a + 6\right)\cdot 19 + \left(15 a^{3} + 12 a^{2} + 8 a + 6\right)\cdot 19^{2} + \left(10 a^{3} + 14 a^{2} + 5 a + 4\right)\cdot 19^{3} + \left(9 a^{3} + a^{2} + a + 14\right)\cdot 19^{4} + \left(17 a^{3} + 4 a^{2} + 7 a + 10\right)\cdot 19^{5} + \left(12 a^{3} + 13 a^{2} + 11 a + 10\right)\cdot 19^{6} + \left(14 a^{3} + 16 a^{2} + 5 a + 9\right)\cdot 19^{7} + \left(13 a^{3} + 18 a^{2} + 6 a + 18\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 7 a^{3} + 12 a^{2} + 9 a + 8 + \left(7 a^{3} + 18 a^{2} + 3\right)\cdot 19 + \left(6 a^{3} + 7 a + 6\right)\cdot 19^{2} + \left(6 a^{3} + a^{2} + 16 a + 15\right)\cdot 19^{3} + \left(11 a^{3} + 11 a^{2} + 14 a + 4\right)\cdot 19^{4} + \left(8 a^{3} + 17 a^{2} + 14 a + 17\right)\cdot 19^{5} + \left(5 a^{3} + 6 a^{2} + 13\right)\cdot 19^{6} + \left(6 a^{3} + 13 a^{2} + 17\right)\cdot 19^{7} + \left(3 a^{3} + 4 a + 13\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 18 a^{3} + 17 a^{2} + 10 a + 4 + \left(5 a^{3} + a^{2} + 14 a + 8\right)\cdot 19 + \left(14 a^{3} + 6 a^{2} + 18 a + 5\right)\cdot 19^{2} + \left(a^{3} + 2 a^{2} + 12 a + 7\right)\cdot 19^{3} + \left(3 a^{3} + 16 a^{2} + 11 a + 8\right)\cdot 19^{4} + \left(2 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 19^{5} + \left(12 a^{3} + 14 a^{2} + 5 a + 14\right)\cdot 19^{6} + \left(14 a^{3} + 5 a^{2} + 11 a + 7\right)\cdot 19^{7} + \left(10 a^{3} + 5 a^{2} + 18 a + 13\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 12 a^{3} + 11 a + 4 + \left(10 a^{2} + a + 15\right)\cdot 19 + \left(9 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 19^{2} + \left(4 a^{3} + 9 a^{2} + 15 a + 3\right)\cdot 19^{3} + \left(3 a^{3} + 4 a + 8\right)\cdot 19^{4} + \left(17 a^{3} + 16 a^{2} + 18 a + 10\right)\cdot 19^{5} + \left(8 a^{3} + 7 a^{2} + 18 a\right)\cdot 19^{6} + \left(17 a^{3} + 11 a^{2} + 10 a + 13\right)\cdot 19^{7} + \left(11 a^{3} + 3 a^{2} + 2 a + 11\right)\cdot 19^{8} +O\left(19^{ 9 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 4 a^{3} + 9 a^{2} + 6 a + 4 + \left(15 a^{3} + 7 a^{2} + 17 a + 14\right)\cdot 19 + \left(12 a^{3} + 7 a^{2} + 9 a + 3\right)\cdot 19^{2} + \left(14 a^{3} + 12 a^{2} + 12 a + 5\right)\cdot 19^{3} + \left(15 a^{3} + 18 a^{2} + 11 a + 6\right)\cdot 19^{4} + \left(14 a^{3} + 16 a^{2} + 4 a + 6\right)\cdot 19^{5} + \left(3 a^{3} + 14 a + 8\right)\cdot 19^{6} + \left(10 a^{3} + 16 a^{2} + a\right)\cdot 19^{7} + \left(3 a^{3} + 6 a^{2} + 5 a + 17\right)\cdot 19^{8} +O\left(19^{ 9 }\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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-2$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)(7,9)(8,10)$ | $0$ | $0$ |
| $3$ | $2$ | $(1,9)(2,10)(3,7)(4,8)(5,11)(6,12)$ | $0$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,4,11)(5,8,10)(7,12,9)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,4,7,10)(2,9,8,3)(5,6,11,12)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,10,7,4)(2,3,8,9)(5,12,11,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $3$ | $4$ | $(1,10,7,4)(2,6,8,12)(3,5,9,11)$ | $0$ | $0$ |
| $3$ | $4$ | $(1,4,7,10)(2,12,8,6)(3,11,9,5)$ | $0$ | $0$ |
| $2$ | $6$ | $(1,9,6,7,3,12)(2,5,4,8,11,10)$ | $1$ | $1$ |
| $2$ | $12$ | $(1,5,9,4,6,8,7,11,3,10,12,2)$ | $-\zeta_{4}$ | $\zeta_{4}$ |
| $2$ | $12$ | $(1,11,9,10,6,2,7,5,3,4,12,8)$ | $\zeta_{4}$ | $-\zeta_{4}$ |