Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_6$ |
| Conductor: | $1083= 3 \cdot 19^{2} $ |
| Artin number field: | Splitting field of $f= x^{18} - 7 x^{17} + 24 x^{16} - 63 x^{15} + 140 x^{14} - 266 x^{13} + 466 x^{12} - 738 x^{11} + 1061 x^{10} - 1410 x^{9} + 1756 x^{8} - 1965 x^{7} + 1884 x^{6} - 1470 x^{5} + 891 x^{4} - 390 x^{3} + 112 x^{2} - 16 x + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 18 a^{5} + 2 a^{3} + 11 a^{2} + 5 a + 19 + \left(23 a^{5} + 21 a^{4} + 2 a^{3} + 11 a^{2} + 11 a + 9\right)\cdot 29 + \left(11 a^{5} + 15 a^{4} + 12 a^{3} + 9 a^{2} + 13 a + 21\right)\cdot 29^{2} + \left(9 a^{5} + 9 a^{4} + 18 a^{2} + 17 a + 4\right)\cdot 29^{3} + \left(20 a^{5} + 8 a^{4} + 13 a^{3} + 12 a^{2} + 25 a + 2\right)\cdot 29^{4} + \left(16 a^{5} + 22 a^{4} + 22 a^{3} + 27 a^{2} + 6 a + 16\right)\cdot 29^{5} + \left(16 a^{5} + 19 a^{4} + 4 a^{3} + 22 a^{2} + 22 a + 5\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 17 a^{5} + 24 a^{4} + 23 a^{3} + 5 a + 20 + \left(6 a^{5} + 12 a^{4} + 2 a^{3} + 16 a^{2} + 3 a + 22\right)\cdot 29 + \left(27 a^{5} + 3 a^{4} + 8 a^{3} + 16 a^{2} + 12 a + 22\right)\cdot 29^{2} + \left(9 a^{5} + 20 a^{4} + 14 a^{3} + 12 a^{2} + 15 a + 21\right)\cdot 29^{3} + \left(19 a^{5} + 25 a^{3} + 13 a^{2} + a + 25\right)\cdot 29^{4} + \left(16 a^{5} + a^{4} + 24 a^{3} + 13 a^{2} + 5 a + 26\right)\cdot 29^{5} + \left(a^{5} + 9 a^{4} + 18 a^{3} + 6 a^{2} + 15 a + 10\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 11 a^{5} + 18 a^{4} + 6 a^{3} + 2 a^{2} + 25 a + 10 + \left(25 a^{5} + 18 a^{4} + 18 a^{3} + 12 a^{2} + 17\right)\cdot 29 + \left(13 a^{5} + 23 a^{4} + 19 a^{3} + 9 a^{2} + 18 a + 17\right)\cdot 29^{2} + \left(24 a^{5} + 4 a^{4} + 4 a^{3} + 15 a^{2} + 14 a + 4\right)\cdot 29^{3} + \left(20 a^{5} + 8 a^{4} + 3 a^{3} + 17 a^{2} + 4 a + 18\right)\cdot 29^{4} + \left(9 a^{5} + 23 a^{4} + 27 a^{3} + 27 a + 21\right)\cdot 29^{5} + \left(16 a^{5} + 27 a^{4} + 21 a^{3} + 26 a^{2} + 13 a + 27\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 17 a^{5} + 25 a^{4} + 23 a^{3} + 4 a^{2} + 26 a + 18 + \left(18 a^{5} + 19 a^{4} + 4 a^{3} + 17 a^{2} + a + 20\right)\cdot 29 + \left(4 a^{5} + 23 a^{3} + 7 a^{2} + 19 a + 7\right)\cdot 29^{2} + \left(5 a^{5} + 10 a^{4} + 26 a^{3} + 11 a^{2} + 5 a + 22\right)\cdot 29^{3} + \left(24 a^{5} + 17 a^{4} + 12 a^{3} + 2 a^{2} + 6 a + 14\right)\cdot 29^{4} + \left(27 a^{5} + 21 a^{4} + 9 a^{3} + 7 a^{2} + 20 a + 27\right)\cdot 29^{5} + \left(8 a^{5} + 2 a^{4} + 20 a^{3} + 11 a^{2} + a + 26\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 9 a^{5} + 5 a^{4} + 28 a^{3} + 22 a^{2} + 4 a + 19 + \left(8 a^{5} + 8 a^{4} + 7 a^{3} + 10 a^{2} + 11 a + 22\right)\cdot 29 + \left(24 a^{4} + 22 a^{3} + 19 a^{2} + 12 a + 8\right)\cdot 29^{2} + \left(23 a^{5} + 3 a^{4} + 20 a^{3} + 19 a^{2} + 19 a + 17\right)\cdot 29^{3} + \left(20 a^{5} + 20 a^{4} + 21 a^{3} + 18 a^{2} + 9 a + 20\right)\cdot 29^{4} + \left(28 a^{5} + 13 a^{4} + 24 a^{3} + 27 a^{2} + 2 a + 18\right)\cdot 29^{5} + \left(15 a^{5} + 21 a^{4} + 10 a^{3} + 24 a^{2} + 3 a + 24\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 8 a^{5} + 3 a^{4} + 11 a^{2} + a + 4 + \left(2 a^{5} + 22 a^{3} + 28 a^{2} + 27 a + 3\right)\cdot 29 + \left(8 a^{5} + 5 a^{4} + 2 a^{3} + 28 a^{2} + 17 a + 7\right)\cdot 29^{2} + \left(9 a^{5} + 25 a^{4} + 20 a^{3} + 25 a + 1\right)\cdot 29^{3} + \left(8 a^{5} + 6 a^{4} + 21 a^{3} + 5 a^{2} + 12 a + 19\right)\cdot 29^{4} + \left(26 a^{5} + 14 a^{4} + 19 a^{3} + 25 a^{2} + 28 a + 5\right)\cdot 29^{5} + \left(28 a^{5} + 26 a^{4} + 26 a^{3} + 27 a^{2} + 7 a + 15\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 15 a^{5} + 18 a^{4} + 14 a^{3} + 15 a^{2} + 13 a + 24 + \left(19 a^{5} + 14 a^{4} + a^{3} + 16 a^{2} + 11 a + 24\right)\cdot 29 + \left(19 a^{5} + 22 a^{4} + 6 a^{3} + 16 a^{2} + 19 a + 28\right)\cdot 29^{2} + \left(3 a^{5} + 15 a^{4} + 18 a^{3} + 13 a^{2} + 18 a + 6\right)\cdot 29^{3} + \left(17 a^{5} + 20 a^{4} + 10 a^{3} + 4 a^{2} + 23 a + 20\right)\cdot 29^{4} + \left(23 a^{5} + 28 a^{4} + 23 a^{3} + 25 a^{2} + 9 a + 14\right)\cdot 29^{5} + \left(4 a^{5} + 2 a^{4} + 28 a^{3} + 2 a^{2} + 23 a + 25\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 23 a^{5} + 17 a^{4} + 12 a^{3} + 20 a^{2} + 14 a + 16 + \left(22 a^{5} + 15 a^{4} + 15 a^{3} + 27 a^{2} + 13 a + 6\right)\cdot 29 + \left(15 a^{5} + 2 a^{4} + 5 a^{3} + 5 a^{2} + 26 a + 7\right)\cdot 29^{2} + \left(23 a^{5} + 23 a^{4} + 12 a^{3} + 28 a^{2} + 26\right)\cdot 29^{3} + \left(10 a^{5} + 17 a^{4} + 5 a^{3} + 24 a^{2} + 5 a + 23\right)\cdot 29^{4} + \left(28 a^{5} + 3 a^{4} + 17 a^{3} + 28 a^{2} + 24 a + 20\right)\cdot 29^{5} + \left(21 a^{5} + 11 a^{4} + 19 a^{3} + 17 a^{2} + 6 a + 21\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 28 a^{5} + 26 a^{4} + 12 a^{3} + 24 a^{2} + 22 a + 20 + \left(18 a^{5} + 4 a^{4} + 4 a^{3} + 14 a^{2} + a\right)\cdot 29 + \left(a^{5} + 19 a^{4} + 5 a^{3} + 18 a^{2} + 26 a + 19\right)\cdot 29^{2} + \left(12 a^{5} + 2 a^{4} + 12 a^{3} + a^{2} + 18 a + 10\right)\cdot 29^{3} + \left(5 a^{5} + 17 a^{4} + 8 a^{3} + 3 a^{2} + 8 a + 17\right)\cdot 29^{4} + \left(25 a^{5} + 14 a^{4} + 5 a^{3} + 14 a^{2} + 27 a + 10\right)\cdot 29^{5} + \left(2 a^{5} + 3 a^{4} + 18 a^{3} + 9 a^{2} + 6 a + 26\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 2 a^{5} + 22 a^{4} + 17 a^{3} + 11 a^{2} + 3 a + 14 + \left(8 a^{5} + 18 a^{4} + 18 a^{3} + 25 a^{2} + 24 a + 8\right)\cdot 29 + \left(21 a^{5} + 15 a^{4} + 23 a^{3} + 5 a^{2} + 16 a + 11\right)\cdot 29^{2} + \left(24 a^{5} + 15 a^{4} + 18 a^{3} + 24 a^{2} + 12 a + 4\right)\cdot 29^{3} + \left(5 a^{5} + 27 a^{4} + 8 a^{3} + a^{2} + 18 a + 5\right)\cdot 29^{4} + \left(11 a^{5} + 15 a^{4} + 4 a^{3} + 20 a + 2\right)\cdot 29^{5} + \left(6 a^{5} + 10 a^{4} + 16 a + 13\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 12 a^{5} + 19 a^{4} + 9 a^{3} + 11 a^{2} + 24 a + 8 + \left(20 a^{5} + 3 a^{4} + 7 a^{3} + 2 a^{2} + 3 a + 14\right)\cdot 29 + \left(2 a^{5} + 8 a^{4} + 10 a^{3} + 24 a^{2} + 10 a + 15\right)\cdot 29^{2} + \left(16 a^{4} + 9 a^{3} + 14 a^{2} + 15 a + 7\right)\cdot 29^{3} + \left(13 a^{5} + 3 a^{4} + 28 a^{3} + 26 a^{2} + 26 a + 20\right)\cdot 29^{4} + \left(13 a^{5} + 25 a^{4} + 22 a^{3} + 3 a^{2} + 10 a + 21\right)\cdot 29^{5} + \left(6 a^{5} + 2 a^{4} + 25 a^{3} + 10 a^{2} + 10 a + 20\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 21 a^{5} + 24 a^{4} + a^{3} + 17 a^{2} + 11 a + 20 + \left(20 a^{5} + 25 a^{4} + 20 a^{3} + 5 a^{2} + 25 a + 23\right)\cdot 29 + \left(12 a^{5} + 21 a^{4} + 11 a^{3} + 8 a^{2} + 25 a + 11\right)\cdot 29^{2} + \left(18 a^{5} + 8 a^{4} + 16 a^{3} + 27 a^{2} + a + 12\right)\cdot 29^{3} + \left(21 a^{5} + 15 a^{4} + 25 a^{3} + 25 a^{2} + 14 a + 28\right)\cdot 29^{4} + \left(23 a^{5} + 4 a^{4} + 21 a^{3} + 15 a^{2} + a + 15\right)\cdot 29^{5} + \left(5 a^{4} + 5 a^{3} + 26 a^{2} + 27 a + 26\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 13 }$ | $=$ | $ 3 a^{5} + 7 a^{4} + 23 a^{3} + a^{2} + 11 a + 17 + \left(9 a^{5} + 20 a^{4} + 18 a^{3} + 2 a^{2} + 3 a + 28\right)\cdot 29 + \left(25 a^{5} + 4 a^{4} + 11 a^{3} + 24 a^{2} + 24 a + 15\right)\cdot 29^{2} + \left(23 a^{5} + 14 a^{4} + 10 a^{3} + 2 a^{2} + 11 a + 1\right)\cdot 29^{3} + \left(19 a^{5} + 22 a^{4} + 17 a^{3} + 2 a^{2} + 4 a + 1\right)\cdot 29^{4} + \left(13 a^{4} + 26 a^{3} + 28 a^{2} + 25 a + 9\right)\cdot 29^{5} + \left(26 a^{5} + 7 a^{4} + 5 a^{3} + 24 a^{2} + 23 a + 8\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 14 }$ | $=$ | $ 9 a^{5} + 28 a^{4} + 23 a^{3} + 9 a^{2} + 9 a + 7 + \left(19 a^{5} + 27 a^{4} + 5 a^{3} + 20 a^{2} + 6 a + 24\right)\cdot 29 + \left(23 a^{5} + 25 a^{4} + 23 a^{3} + 20 a^{2} + a + 27\right)\cdot 29^{2} + \left(4 a^{4} + 9 a^{3} + 10 a^{2} + 20 a + 18\right)\cdot 29^{3} + \left(25 a^{5} + 20 a^{4} + 13 a^{3} + 20 a^{2} + 22 a + 14\right)\cdot 29^{4} + \left(a^{5} + 28 a^{4} + 28 a^{3} + 25 a^{2} + 11 a + 19\right)\cdot 29^{5} + \left(13 a^{5} + 24 a^{4} + 15 a^{3} + 2 a^{2} + 19 a + 23\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 15 }$ | $=$ | $ 20 a^{5} + 19 a^{4} + 4 a^{3} + 22 a^{2} + 16 a + 17 + \left(26 a^{5} + 13 a^{4} + 23 a + 2\right)\cdot 29 + \left(21 a^{5} + 20 a^{4} + 2 a^{3} + 3 a^{2} + 25 a + 11\right)\cdot 29^{2} + \left(26 a^{5} + 11 a^{4} + 11 a^{3} + 18 a^{2} + 13 a + 1\right)\cdot 29^{3} + \left(4 a^{5} + 2 a^{4} + 8 a^{3} + 18 a^{2} + 6 a + 16\right)\cdot 29^{4} + \left(5 a^{5} + 8 a^{4} + 4 a^{3} + 5 a^{2} + 28 a + 2\right)\cdot 29^{5} + \left(13 a^{5} + 6 a^{4} + 23 a^{3} + 5 a^{2} + 10 a + 24\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 16 }$ | $=$ | $ 27 a^{5} + 17 a^{4} + 9 a^{3} + 5 a^{2} + 27 a + 6 + \left(18 a^{5} + 19 a^{4} + 6 a^{3} + 10 a^{2} + 10 a + 8\right)\cdot 29 + \left(9 a^{5} + a^{4} + 10 a^{3} + 19 a^{2} + 12 a + 1\right)\cdot 29^{2} + \left(24 a^{4} + 14 a^{3} + 3 a + 21\right)\cdot 29^{3} + \left(18 a^{5} + 3 a^{4} + a^{3} + 19 a^{2} + 8 a + 22\right)\cdot 29^{4} + \left(27 a^{5} + 28 a^{4} + 15 a^{3} + 5 a^{2} + 23 a + 26\right)\cdot 29^{5} + \left(4 a^{5} + a^{4} + 13 a^{3} + 9 a^{2} + 3 a + 26\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 17 }$ | $=$ | $ 23 a^{5} + 13 a^{4} + 19 a^{3} + 14 a^{2} + 16 a + 14 + \left(25 a^{5} + 23 a^{4} + 12 a^{3} + 3 a^{2} + 20\right)\cdot 29 + \left(11 a^{5} + 5 a^{4} + 2 a^{3} + 13 a^{2} + 14 a + 21\right)\cdot 29^{2} + \left(23 a^{5} + 16 a^{4} + 11 a^{3} + 25 a^{2} + 24 a + 4\right)\cdot 29^{3} + \left(27 a^{5} + 6 a^{4} + 10 a^{3} + 14 a^{2} + a + 16\right)\cdot 29^{4} + \left(4 a^{5} + 4 a^{4} + 7 a^{3} + 25 a^{2} + 2 a + 26\right)\cdot 29^{5} + \left(13 a^{5} + 9 a^{4} + 4 a^{3} + 23 a^{2} + 5 a + 23\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 18 }$ | $=$ | $ 27 a^{5} + 5 a^{4} + 7 a^{3} + 4 a^{2} + 15 + \left(23 a^{5} + 21 a^{4} + 5 a^{3} + 7 a^{2} + 23 a + 2\right)\cdot 29 + \left(28 a^{5} + 10 a^{4} + 3 a^{3} + 10 a^{2} + 23 a + 4\right)\cdot 29^{2} + \left(21 a^{5} + 5 a^{4} + a^{3} + 15 a^{2} + 20 a + 15\right)\cdot 29^{3} + \left(6 a^{5} + 13 a^{4} + 25 a^{3} + 2 a + 3\right)\cdot 29^{4} + \left(27 a^{5} + 18 a^{4} + 13 a^{3} + 10 a^{2} + 15 a + 3\right)\cdot 29^{5} + \left(9 a^{4} + 25 a^{3} + 8 a^{2} + 13 a + 25\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
Generators of the action on the roots $r_1, \ldots, r_{ 18 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 18 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,10)(2,7)(3,15)(4,13)(5,14)(6,12)(8,18)(9,17)(11,16)$ | $-2$ | $-2$ |
| $3$ | $2$ | $(1,9)(2,4)(3,15)(5,12)(6,14)(7,13)(8,18)(10,17)(11,16)$ | $0$ | $0$ |
| $3$ | $2$ | $(1,18)(3,5)(4,11)(8,10)(13,16)(14,15)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,5,4)(2,9,12)(3,11,18)(6,7,17)(8,15,16)(10,14,13)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,5)(2,12,9)(3,18,11)(6,17,7)(8,16,15)(10,13,14)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,18,17)(2,13,16)(3,6,5)(4,11,7)(8,9,10)(12,14,15)$ | $-1$ | $-1$ |
| $2$ | $3$ | $(1,6,11)(2,8,14)(3,4,17)(5,7,18)(9,15,13)(10,12,16)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,11,6)(2,14,8)(3,17,4)(5,18,7)(9,13,15)(10,16,12)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,13,5,10,4,14)(2,6,9,7,12,17)(3,8,11,15,18,16)$ | $2 \zeta_{3} + 2$ | $-2 \zeta_{3}$ |
| $1$ | $6$ | $(1,14,4,10,5,13)(2,17,12,7,9,6)(3,16,18,15,11,8)$ | $-2 \zeta_{3}$ | $2 \zeta_{3} + 2$ |
| $2$ | $6$ | $(1,9,18,10,17,8)(2,11,13,7,16,4)(3,14,6,15,5,12)$ | $1$ | $1$ |
| $2$ | $6$ | $(1,16,6,10,11,12)(2,5,8,7,14,18)(3,9,4,15,17,13)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,12,11,10,6,16)(2,18,14,7,8,5)(3,13,17,15,4,9)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $3$ | $6$ | $(1,2,5,9,4,12)(3,8,11,15,18,16)(6,10,7,14,17,13)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,12,4,9,5,2)(3,16,18,15,11,8)(6,13,17,14,7,10)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,4,5)(2,15,9,16,12,8)(3,17,11,6,18,7)(10,13,14)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,5,4)(2,8,12,16,9,15)(3,7,18,6,11,17)(10,14,13)$ | $0$ | $0$ |