Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $635= 5 \cdot 127 $ |
| Artin number field: | Splitting field of $f= x^{12} - x^{11} + 43 x^{10} - 165 x^{9} + 2051 x^{8} + 13810 x^{7} + 85532 x^{6} + 330408 x^{5} + 2157136 x^{4} + 4877440 x^{3} + 10777600 x^{2} + 21504000 x + 40960000 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $C_{12}$ |
| Parity: | Odd |
| Corresponding Dirichlet character: | \(\chi_{635}(273,\cdot)\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{4} + 8 x^{2} + 40 x + 5 $
Roots:
| $r_{ 1 }$ | $=$ | $ 10 a^{3} + 25 a^{2} + 35 a + 13 + \left(20 a^{3} + 16 a^{2} + 38 a + 17\right)\cdot 47 + \left(24 a^{3} + 16 a^{2} + 44 a + 39\right)\cdot 47^{2} + \left(15 a^{3} + 40 a^{2} + 43 a + 33\right)\cdot 47^{3} + \left(29 a^{3} + 6 a^{2} + 7 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 7 a^{3} + 10 a^{2} + 10 a + 4 + \left(41 a^{3} + 40 a^{2} + 11 a + 35\right)\cdot 47 + \left(40 a^{3} + 24 a^{2} + 10 a + 2\right)\cdot 47^{2} + \left(46 a^{3} + 16 a^{2} + 27 a + 33\right)\cdot 47^{3} + \left(24 a^{3} + 17 a^{2} + 30 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 19 a^{3} + 18 a^{2} + 9 a + 27 + \left(13 a^{3} + 9 a^{2} + 21 a + 35\right)\cdot 47 + \left(45 a^{3} + 10 a^{2} + 19 a + 12\right)\cdot 47^{2} + \left(4 a^{3} + 29 a^{2} + 29 a + 45\right)\cdot 47^{3} + \left(32 a^{3} + 34 a^{2} + 4 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ a^{3} + 43 a^{2} + 45 a + 41 + \left(17 a^{3} + 11 a^{2} + 17 a + 7\right)\cdot 47 + \left(31 a^{3} + 27 a^{2} + 2 a + 11\right)\cdot 47^{2} + \left(a^{3} + 10 a^{2} + 30 a + 1\right)\cdot 47^{3} + \left(39 a^{3} + 39 a^{2} + 16 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 9 a^{3} + 12 a^{2} + 43 a + 32 + \left(31 a^{3} + 17 a^{2} + 26 a + 36\right)\cdot 47 + \left(32 a^{3} + 27 a^{2} + 5 a + 31\right)\cdot 47^{2} + \left(21 a^{3} + 34 a^{2} + 32 a + 4\right)\cdot 47^{3} + \left(25 a^{2} + 14 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 45 a^{3} + 13 a^{2} + 27 a + 19 + \left(20 a^{3} + 6 a + 32\right)\cdot 47 + \left(12 a^{3} + 3 a^{2} + 21 a + 4\right)\cdot 47^{2} + \left(46 a^{3} + 10 a^{2} + 34 a + 21\right)\cdot 47^{3} + \left(2 a^{2} + 12 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 17 a^{3} + 26 a^{2} + 13 a + 39 + \left(9 a^{3} + 6 a^{2} + 19 a + 27\right)\cdot 47 + \left(43 a^{3} + 45 a^{2} + 43 a + 12\right)\cdot 47^{2} + \left(12 a^{3} + 33 a^{2} + 25 a + 24\right)\cdot 47^{3} + \left(30 a^{3} + 40 a^{2} + 2 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 13 a^{3} + 33 a^{2} + 36 a + 41 + \left(23 a^{3} + 30 a^{2} + 24 a + 24\right)\cdot 47 + \left(32 a^{3} + 7 a^{2} + 42 a + 11\right)\cdot 47^{2} + \left(18 a^{3} + 3 a^{2} + 43 a + 27\right)\cdot 47^{3} + \left(9 a^{3} + 29 a^{2} + 5 a + 29\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 25 a^{3} + 39 a^{2} + 17 a + 9 + \left(22 a^{3} + 41 a^{2} + 22 a + 16\right)\cdot 47 + \left(24 a^{3} + 39 a^{2} + 12 a + 24\right)\cdot 47^{2} + \left(a^{3} + 10 a^{2} + 22 a + 9\right)\cdot 47^{3} + \left(35 a^{3} + 16 a^{2} + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 31 a^{3} + 7 a^{2} + 38 a + 45 + \left(19 a^{3} + 44 a + 39\right)\cdot 47 + \left(17 a^{3} + 37 a^{2} + a + 7\right)\cdot 47^{2} + \left(12 a^{3} + 44 a^{2} + a + 36\right)\cdot 47^{3} + \left(19 a^{3} + 12 a^{2} + 5 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 41 a^{3} + 25 a^{2} + 25 a + 10 + \left(26 a^{3} + 25 a^{2} + 23 a + 34\right)\cdot 47 + \left(38 a^{3} + 16 a^{2} + 9 a + 24\right)\cdot 47^{2} + \left(18 a^{3} + 19 a^{2} + 10 a + 45\right)\cdot 47^{3} + \left(26 a^{3} + 17 a^{2} + 27 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 17 a^{3} + 31 a^{2} + 31 a + 3 + \left(36 a^{3} + 34 a^{2} + 24 a + 21\right)\cdot 47 + \left(32 a^{3} + 26 a^{2} + 21 a + 4\right)\cdot 47^{2} + \left(33 a^{3} + 28 a^{2} + 28 a\right)\cdot 47^{3} + \left(34 a^{3} + 39 a^{2} + 12 a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-1$ |
| $1$ | $3$ | $(1,9,10)(2,11,12)(3,4,7)(5,6,8)$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $3$ | $(1,10,9)(2,12,11)(3,7,4)(5,8,6)$ | $-\zeta_{12}^{2}$ |
| $1$ | $4$ | $(1,8,7,2)(3,11,9,5)(4,12,10,6)$ | $\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,2,7,8)(3,5,9,11)(4,6,10,12)$ | $-\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,4,9,7,10,3)(2,6,11,8,12,5)$ | $\zeta_{12}^{2}$ |
| $1$ | $6$ | $(1,3,10,7,9,4)(2,5,12,8,11,6)$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $12$ | $(1,11,4,8,9,12,7,5,10,2,3,6)$ | $\zeta_{12}$ |
| $1$ | $12$ | $(1,12,3,8,10,11,7,6,9,2,4,5)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,5,4,2,9,6,7,11,10,8,3,12)$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,6,3,2,10,5,7,12,9,8,4,11)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |