Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_4$ |
| Conductor: | $676= 2^{2} \cdot 13^{2} $ |
| Artin number field: | Splitting field of $f= x^{12} - 3 x^{11} + 9 x^{10} - 27 x^{9} + 55 x^{8} - 100 x^{7} + 157 x^{6} - 185 x^{5} + 152 x^{4} - 40 x^{3} - 36 x^{2} - 9 x + 27 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $S_3 \times C_4$ |
| Parity: | Odd |
| Determinant: | 1.2e2_13.2t1.1c1 |
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:
| $r_{ 1 }$ | $=$ | $ 12 a^{3} + 5 a^{2} + 18 a + 1 + \left(7 a^{2} + 16 a + 1\right)\cdot 19 + \left(17 a^{2} + 7 a + 15\right)\cdot 19^{2} + \left(11 a^{3} + 12 a^{2} + 7 a + 14\right)\cdot 19^{3} + \left(16 a^{3} + 17 a^{2} + 2 a\right)\cdot 19^{4} + \left(8 a^{3} + 13 a^{2} + 9 a + 9\right)\cdot 19^{5} + \left(4 a^{3} + 7 a^{2} + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 15 a^{3} + 5 a^{2} + 4 + \left(8 a^{3} + 17 a^{2} + 18 a + 3\right)\cdot 19 + \left(a^{3} + 6 a^{2} + 13 a + 5\right)\cdot 19^{2} + \left(10 a^{3} + 17 a^{2} + 3 a + 4\right)\cdot 19^{3} + \left(13 a^{3} + 15 a^{2} + 13 a + 14\right)\cdot 19^{4} + \left(4 a^{3} + 15 a^{2} + 5 a + 6\right)\cdot 19^{5} + \left(5 a^{3} + 18 a^{2} + 10\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 5 a^{3} + 6 a^{2} + 5 a + 8 + \left(8 a^{3} + 18 a^{2} + 11 a + 9\right)\cdot 19 + \left(15 a^{3} + 6 a^{2} + 6\right)\cdot 19^{2} + \left(12 a^{3} + 15 a^{2} + 5 a + 15\right)\cdot 19^{3} + \left(12 a^{3} + 16 a^{2} + 3 a + 12\right)\cdot 19^{4} + \left(15 a^{3} + 4 a^{2} + a\right)\cdot 19^{5} + \left(15 a^{3} + 12 a^{2} + 17 a + 15\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ a^{3} + 7 a + 5 + \left(a^{3} + 17 a^{2} + 12 a + 14\right)\cdot 19 + \left(7 a^{3} + 12 a^{2} + 2 a + 6\right)\cdot 19^{2} + \left(a^{3} + 18 a^{2} + 2 a + 12\right)\cdot 19^{3} + \left(15 a^{3} + 9 a^{2} + 5 a + 18\right)\cdot 19^{4} + \left(11 a^{3} + 6 a^{2} + 12 a + 1\right)\cdot 19^{5} + \left(18 a^{3} + 9 a^{2} + 5 a + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 3 a^{3} + 12 a^{2} + 4 + \left(14 a^{3} + 8 a^{2}\right)\cdot 19 + \left(a^{3} + 13 a^{2} + 8 a + 11\right)\cdot 19^{2} + \left(17 a^{3} + 4 a^{2} + a + 12\right)\cdot 19^{3} + \left(15 a^{3} + 10 a^{2} + 13 a + 4\right)\cdot 19^{4} + \left(10 a^{3} + 2 a^{2} + 17 a + 4\right)\cdot 19^{5} + \left(7 a^{3} + 14 a^{2} + 8 a + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 11 a^{3} + 4 a^{2} + a + 8 + \left(a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 19 + \left(6 a^{3} + 17 a^{2} + 8 a + 15\right)\cdot 19^{2} + \left(8 a^{2} + 11 a\right)\cdot 19^{3} + \left(2 a^{3} + 17 a^{2} + 16 a + 11\right)\cdot 19^{4} + \left(10 a^{3} + 4 a^{2} + 7 a + 16\right)\cdot 19^{5} + \left(17 a^{3} + 8 a^{2} + 5 a + 10\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 9 a^{3} + 3 a^{2} + 4 a + 17 + \left(8 a^{3} + 10 a^{2} + 13 a + 6\right)\cdot 19 + \left(15 a^{3} + 7 a^{2} + 8 a + 18\right)\cdot 19^{2} + \left(13 a^{3} + 10 a^{2} + 18 a + 6\right)\cdot 19^{3} + \left(18 a^{3} + 16 a^{2} + 8 a + 17\right)\cdot 19^{4} + \left(5 a^{3} + 5 a^{2}\right)\cdot 19^{5} + \left(14 a^{3} + a^{2} + 13 a + 7\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 4 a^{3} + 12 a^{2} + 10 a + 17 + \left(14 a^{3} + 11 a^{2} + 18 a + 17\right)\cdot 19 + \left(5 a^{3} + 3 a^{2} + 11 a\right)\cdot 19^{2} + \left(10 a^{3} + 8 a^{2} + 11 a + 17\right)\cdot 19^{3} + \left(9 a^{3} + 9 a^{2} + 18 a + 17\right)\cdot 19^{4} + \left(15 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 19^{5} + \left(15 a^{3} + 11 a^{2} + 12 a + 16\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 5 a^{3} + a^{2} + 12 a + \left(16 a^{3} + 18 a^{2} + 16 a + 8\right)\cdot 19 + \left(5 a^{3} + a^{2} + 12 a + 14\right)\cdot 19^{2} + \left(5 a^{2} + 18 a + 2\right)\cdot 19^{3} + \left(3 a^{3} + 13 a^{2} + 7 a + 1\right)\cdot 19^{4} + \left(4 a^{3} + a^{2} + 16 a + 10\right)\cdot 19^{5} + \left(15 a^{3} + 8 a^{2} + 16 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 7 a^{3} + 4 a^{2} + 13 a + 13 + \left(16 a^{2} + 5 a + 8\right)\cdot 19 + \left(15 a^{3} + 6 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(14 a^{3} + 15 a^{2} + 17 a + 12\right)\cdot 19^{3} + \left(9 a^{3} + 6 a^{2} + 4 a + 2\right)\cdot 19^{4} + \left(7 a^{3} + 12 a^{2} + 4 a + 12\right)\cdot 19^{5} + \left(18 a^{3} + 17 a^{2} + 15 a + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 16 a^{3} + 11 a^{2} + 9 a + 2 + \left(8 a^{3} + 3 a^{2} + 14 a + 8\right)\cdot 19 + \left(15 a^{3} + 18 a + 6\right)\cdot 19^{2} + \left(11 a^{3} + 15 a^{2} + 9 a + 9\right)\cdot 19^{3} + \left(6 a^{3} + 12 a^{2} + 2 a + 13\right)\cdot 19^{4} + \left(11 a^{3} + 11 a^{2} + 16 a + 17\right)\cdot 19^{5} + \left(18 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 7 a^{3} + 13 a^{2} + 16 a + \left(12 a^{3} + 18 a^{2} + 2 a + 5\right)\cdot 19 + \left(5 a^{3} + 18 a^{2} + 5 a + 1\right)\cdot 19^{2} + \left(10 a^{3} + 6 a + 5\right)\cdot 19^{3} + \left(9 a^{3} + 5 a^{2} + 17 a + 18\right)\cdot 19^{4} + \left(7 a^{3} + 5 a^{2} + 15 a + 7\right)\cdot 19^{5} + \left(18 a^{3} + 4 a^{2} + 18 a + 11\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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,10)(4,11)(5,9)(8,12)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,11)(4,10)(5,9)(6,7)(8,12)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,7)(3,4)(10,11)$ | $0$ |
| $2$ | $3$ | $(1,6,12)(2,8,7)(3,4,9)(5,10,11)$ | $-1$ |
| $1$ | $4$ | $(1,3,7,10)(2,11,6,4)(5,12,9,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,10,7,3)(2,4,6,11)(5,8,9,12)$ | $2 \zeta_{4}$ |
| $3$ | $4$ | $(1,9,7,5)(2,11,6,4)(3,8,10,12)$ | $0$ |
| $3$ | $4$ | $(1,5,7,9)(2,4,6,11)(3,12,10,8)$ | $0$ |
| $2$ | $6$ | $(1,8,6,7,12,2)(3,5,4,10,9,11)$ | $1$ |
| $2$ | $12$ | $(1,11,8,3,6,5,7,4,12,10,2,9)$ | $-\zeta_{4}$ |
| $2$ | $12$ | $(1,4,8,10,6,9,7,11,12,3,2,5)$ | $\zeta_{4}$ |