Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_3$ |
| Conductor: | $961= 31^{2} $ |
| Artin number field: | Splitting field of $f= x^{18} - 6 x^{17} + 19 x^{16} - 34 x^{15} + 42 x^{14} - 34 x^{13} + 20 x^{12} + 20 x^{11} - 97 x^{10} + 148 x^{9} - 131 x^{8} - 30 x^{7} + 251 x^{6} - 76 x^{5} - 53 x^{4} + 8 x^{3} + 11 x^{2} + 4 x + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $S_3\times C_3$ |
| Parity: | Odd |
| Determinant: | 1.31.6t1.1c1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 19 a^{5} + 24 a^{4} + 34 a^{3} + 34 a^{2} + 13 a + 12 + \left(17 a^{5} + 30 a^{4} + 10 a^{3} + 33 a^{2} + 7 a + 32\right)\cdot 37 + \left(35 a^{5} + 30 a^{4} + 8 a^{3} + 6 a^{2} + 27 a + 28\right)\cdot 37^{2} + \left(21 a^{5} + 6 a^{4} + 18 a^{3} + 11 a^{2} + 30 a + 22\right)\cdot 37^{3} + \left(5 a^{5} + 14 a^{4} + 5 a^{3} + 8 a^{2} + 13 a + 8\right)\cdot 37^{4} + \left(12 a^{5} + 3 a^{4} + 15 a^{3} + 20 a^{2} + 11 a + 35\right)\cdot 37^{5} + \left(23 a^{5} + 35 a^{4} + 9 a^{3} + 28 a^{2} + 28 a + 8\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 3 a^{5} + 30 a^{4} + 6 a^{3} + 19 a^{2} + 9 a + 20 + \left(14 a^{5} + 10 a^{4} + 19 a^{3} + 9 a^{2} + 5 a + 7\right)\cdot 37 + \left(5 a^{5} + 34 a^{4} + 11 a^{3} + 15 a^{2} + 29 a + 22\right)\cdot 37^{2} + \left(12 a^{5} + 34 a^{4} + 30 a^{3} + 3 a^{2} + 32 a + 27\right)\cdot 37^{3} + \left(11 a^{5} + 7 a^{4} + 5 a^{3} + 25 a^{2} + 35 a + 27\right)\cdot 37^{4} + \left(31 a^{5} + 8 a^{4} + 32 a^{3} + 4 a^{2} + 31 a + 21\right)\cdot 37^{5} + \left(a^{5} + 21 a^{4} + 36 a^{3} + 24 a^{2} + 7 a + 16\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 3 a^{5} + 30 a^{4} + 6 a^{3} + 19 a^{2} + 9 a + 32 + \left(14 a^{5} + 10 a^{4} + 19 a^{3} + 9 a^{2} + 5 a + 18\right)\cdot 37 + \left(5 a^{5} + 34 a^{4} + 11 a^{3} + 15 a^{2} + 29 a + 6\right)\cdot 37^{2} + \left(12 a^{5} + 34 a^{4} + 30 a^{3} + 3 a^{2} + 32 a + 26\right)\cdot 37^{3} + \left(11 a^{5} + 7 a^{4} + 5 a^{3} + 25 a^{2} + 35 a + 7\right)\cdot 37^{4} + \left(31 a^{5} + 8 a^{4} + 32 a^{3} + 4 a^{2} + 31 a + 24\right)\cdot 37^{5} + \left(a^{5} + 21 a^{4} + 36 a^{3} + 24 a^{2} + 7 a + 11\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 20 a^{5} + 30 a^{3} + 20 a^{2} + 17 a + 14 + \left(2 a^{5} + 35 a^{4} + 36 a^{3} + 14 a^{2} + 8 a + 36\right)\cdot 37 + \left(3 a^{5} + 28 a^{4} + 5 a^{3} + 33 a^{2} + 14 a + 5\right)\cdot 37^{2} + \left(35 a^{5} + 8 a^{4} + 29 a^{3} + 6 a^{2} + 34 a + 17\right)\cdot 37^{3} + \left(19 a^{5} + 4 a^{4} + 17 a^{3} + 24 a^{2} + 7 a + 13\right)\cdot 37^{4} + \left(10 a^{5} + 33 a^{4} + 28 a^{3} + 30 a^{2} + 27 a + 5\right)\cdot 37^{5} + \left(6 a^{4} + 21 a^{3} + 33 a^{2} + 32 a + 24\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 27 a^{5} + 3 a^{4} + 15 a^{3} + 5 a^{2} + 29 a + 27 + \left(17 a^{5} + 20 a^{4} + 25 a^{3} + 23 a^{2} + 25 a + 28\right)\cdot 37 + \left(36 a^{5} + 27 a^{4} + 28 a^{3} + 34 a^{2} + 9 a + 19\right)\cdot 37^{2} + \left(23 a^{5} + 27 a^{4} + 26 a^{3} + 34 a^{2} + 23 a + 7\right)\cdot 37^{3} + \left(20 a^{5} + 23 a^{4} + 2 a^{3} + 21 a^{2} + 25 a + 35\right)\cdot 37^{4} + \left(10 a^{5} + 7 a^{4} + 27 a^{3} + 20 a^{2} + 6 a + 11\right)\cdot 37^{5} + \left(18 a^{5} + 31 a^{4} + 11 a^{3} + 22 a^{2} + 36 a + 30\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 36 a^{5} + 31 a^{4} + 21 a^{3} + 35 a^{2} + 21 a + 31 + \left(5 a^{5} + 17 a^{4} + 28 a^{3} + a^{2} + 10 a + 12\right)\cdot 37 + \left(33 a^{5} + 25 a^{4} + 2 a^{3} + 25 a^{2} + 12 a + 28\right)\cdot 37^{2} + \left(7 a^{5} + 17 a^{4} + 17 a^{3} + 14 a^{2} + 32 a + 13\right)\cdot 37^{3} + \left(14 a^{5} + 10 a^{4} + 17 a^{3} + 7 a^{2} + 22 a + 14\right)\cdot 37^{4} + \left(28 a^{5} + 28 a^{4} + 11 a^{3} + 9 a^{2} + 6 a + 16\right)\cdot 37^{5} + \left(21 a^{5} + 20 a^{4} + 31 a^{3} + a^{2} + 16 a + 21\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 36 a^{5} + 31 a^{4} + 21 a^{3} + 35 a^{2} + 21 a + 6 + \left(5 a^{5} + 17 a^{4} + 28 a^{3} + a^{2} + 10 a + 24\right)\cdot 37 + \left(33 a^{5} + 25 a^{4} + 2 a^{3} + 25 a^{2} + 12 a + 12\right)\cdot 37^{2} + \left(7 a^{5} + 17 a^{4} + 17 a^{3} + 14 a^{2} + 32 a + 12\right)\cdot 37^{3} + \left(14 a^{5} + 10 a^{4} + 17 a^{3} + 7 a^{2} + 22 a + 31\right)\cdot 37^{4} + \left(28 a^{5} + 28 a^{4} + 11 a^{3} + 9 a^{2} + 6 a + 18\right)\cdot 37^{5} + \left(21 a^{5} + 20 a^{4} + 31 a^{3} + a^{2} + 16 a + 16\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 7 a^{5} + 10 a^{4} + 2 a^{3} + 6 a^{2} + 21 + \left(6 a^{5} + 7 a^{4} + 6 a^{3} + 28 a^{2} + 29 a + 20\right)\cdot 37 + \left(34 a^{5} + 22 a^{4} + 23 a^{3} + 15 a^{2} + 31 a + 3\right)\cdot 37^{2} + \left(9 a^{5} + a^{4} + 25 a^{3} + a^{2} + 24 a + 34\right)\cdot 37^{3} + \left(29 a^{5} + 20 a^{4} + 14 a^{3} + 21 a^{2} + 34 a + 20\right)\cdot 37^{4} + \left(26 a^{5} + 32 a^{4} + 23 a^{3} + 9 a^{2} + a + 32\right)\cdot 37^{5} + \left(16 a^{5} + 16 a^{4} + 33 a^{3} + 32 a^{2} + 24 a\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 7 a^{5} + 10 a^{4} + 2 a^{3} + 6 a^{2} + 9 + \left(6 a^{5} + 7 a^{4} + 6 a^{3} + 28 a^{2} + 29 a + 9\right)\cdot 37 + \left(34 a^{5} + 22 a^{4} + 23 a^{3} + 15 a^{2} + 31 a + 19\right)\cdot 37^{2} + \left(9 a^{5} + a^{4} + 25 a^{3} + a^{2} + 24 a + 35\right)\cdot 37^{3} + \left(29 a^{5} + 20 a^{4} + 14 a^{3} + 21 a^{2} + 34 a + 3\right)\cdot 37^{4} + \left(26 a^{5} + 32 a^{4} + 23 a^{3} + 9 a^{2} + a + 30\right)\cdot 37^{5} + \left(16 a^{5} + 16 a^{4} + 33 a^{3} + 32 a^{2} + 24 a + 5\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 9 a^{5} + 18 a^{3} + 13 a^{2} + 13 a + 10 + \left(2 a^{5} + 31 a^{4} + 30 a^{3} + 18 a^{2} + 19 a + 6\right)\cdot 37 + \left(12 a^{5} + 8 a^{4} + 26 a^{3} + 15 a + 5\right)\cdot 37^{2} + \left(22 a^{5} + 8 a^{4} + 9 a^{3} + 27 a^{2} + 21 a + 4\right)\cdot 37^{3} + \left(19 a^{5} + 23 a^{4} + 24 a^{3} + 30 a^{2} + 27 a + 33\right)\cdot 37^{4} + \left(25 a^{5} + 22 a^{4} + 8 a^{2} + 27 a\right)\cdot 37^{5} + \left(35 a^{5} + 25 a^{4} + 9 a^{3} + a^{2} + 4 a + 33\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 11 a^{5} + 3 a^{4} + a^{3} + 34 a^{2} + 24 a + 5 + \left(13 a^{5} + 36 a^{4} + 20 a^{3} + 11 a^{2} + 21\right)\cdot 37 + \left(4 a^{5} + 20 a^{4} + 5 a^{3} + 14 a^{2} + 15 a + 4\right)\cdot 37^{2} + \left(5 a^{5} + 28 a^{4} + 30 a^{3} + 24 a^{2} + 18 a + 17\right)\cdot 37^{3} + \left(2 a^{5} + 2 a^{4} + 16 a^{3} + 7 a^{2} + 25 a + 7\right)\cdot 37^{4} + \left(35 a^{5} + a^{4} + 35 a^{3} + 7 a^{2} + 23 a + 6\right)\cdot 37^{5} + \left(33 a^{5} + 22 a^{4} + 30 a^{3} + 13 a^{2} + 21 a + 27\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 11 a^{5} + 3 a^{4} + a^{3} + 34 a^{2} + 24 a + 17 + \left(13 a^{5} + 36 a^{4} + 20 a^{3} + 11 a^{2} + 32\right)\cdot 37 + \left(4 a^{5} + 20 a^{4} + 5 a^{3} + 14 a^{2} + 15 a + 25\right)\cdot 37^{2} + \left(5 a^{5} + 28 a^{4} + 30 a^{3} + 24 a^{2} + 18 a + 15\right)\cdot 37^{3} + \left(2 a^{5} + 2 a^{4} + 16 a^{3} + 7 a^{2} + 25 a + 24\right)\cdot 37^{4} + \left(35 a^{5} + a^{4} + 35 a^{3} + 7 a^{2} + 23 a + 8\right)\cdot 37^{5} + \left(33 a^{5} + 22 a^{4} + 30 a^{3} + 13 a^{2} + 21 a + 22\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 13 }$ | $=$ | $ 31 a^{5} + 33 a^{4} + 14 a^{3} + 33 a^{2} + 16 a + 23 + \left(24 a^{5} + 11 a^{4} + 2 a^{3} + 6 a^{2} + 34 a + 3\right)\cdot 37 + \left(6 a^{5} + 26 a^{4} + 11 a^{3} + 33 a^{2} + 29 a + 5\right)\cdot 37^{2} + \left(19 a^{5} + 17 a^{4} + 31 a^{3} + 20 a^{2} + 16 a + 26\right)\cdot 37^{3} + \left(30 a^{5} + 6 a^{4} + 4 a^{3} + 8 a^{2} + 16 a + 1\right)\cdot 37^{4} + \left(18 a^{5} + 13 a^{4} + 2 a^{3} + 18 a^{2} + 28 a + 25\right)\cdot 37^{5} + \left(35 a^{5} + 36 a^{4} + 9 a^{3} + 3 a^{2} + 33 a + 14\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 14 }$ | $=$ | $ 5 a^{5} + 14 a^{4} + 6 a^{2} + 23 a + 27 + \left(9 a^{5} + 19 a^{4} + 5 a^{3} + 14 a^{2} + 15 a + 3\right)\cdot 37 + \left(17 a^{5} + 25 a^{4} + 30 a^{3} + 2 a^{2} + 14 a + 9\right)\cdot 37^{2} + \left(25 a^{5} + 4 a^{4} + 32 a^{3} + 10 a^{2} + 21 a + 33\right)\cdot 37^{3} + \left(14 a^{5} + 2 a^{4} + 18 a^{3} + 17 a^{2} + 19 a + 18\right)\cdot 37^{4} + \left(33 a^{5} + 31 a^{4} + 12 a^{2} + 9 a + 32\right)\cdot 37^{5} + \left(34 a^{5} + 12 a^{4} + 13 a^{3} + 21 a^{2} + 12 a + 36\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 15 }$ | $=$ | $ 29 a^{5} + 30 a^{4} + 31 a^{3} + 12 a^{2} + 5 a + 28 + \left(13 a^{5} + 6 a^{4} + 12 a^{3} + 13 a^{2} + 16 a + 25\right)\cdot 37 + \left(14 a^{5} + 14 a^{4} + 32 a^{3} + 19 a^{2} + 30 a + 5\right)\cdot 37^{2} + \left(36 a^{5} + 34 a^{4} + 10 a^{3} + 23 a^{2} + 19 a + 13\right)\cdot 37^{3} + \left(10 a^{5} + 26 a^{4} + 12 a^{3} + 31 a^{2} + 18 a + 27\right)\cdot 37^{4} + \left(9 a^{5} + 34 a^{4} + 4 a^{3} + 19 a^{2} + 32 a + 19\right)\cdot 37^{5} + \left(2 a^{4} + 24 a^{3} + 28 a^{2} + 16 a + 20\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 16 }$ | $=$ | $ 29 a^{5} + 30 a^{4} + 31 a^{3} + 12 a^{2} + 5 a + 16 + \left(13 a^{5} + 6 a^{4} + 12 a^{3} + 13 a^{2} + 16 a + 14\right)\cdot 37 + \left(14 a^{5} + 14 a^{4} + 32 a^{3} + 19 a^{2} + 30 a + 21\right)\cdot 37^{2} + \left(36 a^{5} + 34 a^{4} + 10 a^{3} + 23 a^{2} + 19 a + 14\right)\cdot 37^{3} + \left(10 a^{5} + 26 a^{4} + 12 a^{3} + 31 a^{2} + 18 a + 10\right)\cdot 37^{4} + \left(9 a^{5} + 34 a^{4} + 4 a^{3} + 19 a^{2} + 32 a + 17\right)\cdot 37^{5} + \left(2 a^{4} + 24 a^{3} + 28 a^{2} + 16 a + 25\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 17 }$ | $=$ | $ 25 a^{5} + 7 a^{4} + 13 a^{3} + 5 a^{2} + 15 a + 8 + \left(20 a^{5} + 32 a^{4} + 24 a^{3} + 9 a^{2} + 12 a + 23\right)\cdot 37 + \left(19 a^{5} + 30 a^{4} + 35 a^{3} + 21 a^{2} + 29 a + 9\right)\cdot 37^{2} + \left(2 a^{5} + 30 a^{4} + 33 a^{3} + 6 a^{2} + 19 a + 5\right)\cdot 37^{3} + \left(6 a^{5} + 5 a^{4} + 6 a^{3} + 18 a^{2} + 10 a + 13\right)\cdot 37^{4} + \left(17 a^{5} + 6 a^{4} + 4 a^{3} + 23 a^{2} + 14 a + 14\right)\cdot 37^{5} + \left(36 a^{5} + 27 a^{4} + 28 a^{3} + 11 a^{2} + 24 a + 24\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 18 }$ | $=$ | $ 25 a^{5} + 7 a^{4} + 13 a^{3} + 5 a^{2} + 15 a + 33 + \left(20 a^{5} + 32 a^{4} + 24 a^{3} + 9 a^{2} + 12 a + 11\right)\cdot 37 + \left(19 a^{5} + 30 a^{4} + 35 a^{3} + 21 a^{2} + 29 a + 25\right)\cdot 37^{2} + \left(2 a^{5} + 30 a^{4} + 33 a^{3} + 6 a^{2} + 19 a + 6\right)\cdot 37^{3} + \left(6 a^{5} + 5 a^{4} + 6 a^{3} + 18 a^{2} + 10 a + 33\right)\cdot 37^{4} + \left(17 a^{5} + 6 a^{4} + 4 a^{3} + 23 a^{2} + 14 a + 11\right)\cdot 37^{5} + \left(36 a^{5} + 27 a^{4} + 28 a^{3} + 11 a^{2} + 24 a + 29\right)\cdot 37^{6} +O\left(37^{ 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 value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,18)(2,5)(3,9)(4,8)(6,17)(7,14)(10,12)(11,15)(13,16)$ | $0$ |
| $1$ | $3$ | $(1,13,5)(2,18,16)(3,17,15)(4,14,10)(6,11,9)(7,12,8)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,13)(2,16,18)(3,15,17)(4,10,14)(6,9,11)(7,8,12)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,7,6)(2,17,10)(3,14,16)(4,18,15)(5,8,9)(11,13,12)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,7)(2,10,17)(3,16,14)(4,15,18)(5,9,8)(11,12,13)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,12,9)(2,15,14)(3,10,18)(4,16,17)(5,7,11)(6,13,8)$ | $-1$ |
| $3$ | $6$ | $(1,2,13,18,5,16)(3,11,17,9,15,6)(4,12,14,8,10,7)$ | $0$ |
| $3$ | $6$ | $(1,16,5,18,13,2)(3,6,15,9,17,11)(4,7,10,8,14,12)$ | $0$ |