Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $208= 2^{4} \cdot 13 $ |
| Artin number field: | Splitting field of $f= x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $C_{12}$ |
| Parity: | Even |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{4} + 9 x^{2} + 38 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 41 a^{3} + 31 a^{2} + 14 a + 43 + \left(25 a^{3} + 20 a + 30\right)\cdot 53 + \left(46 a^{3} + 31 a^{2} + 10 a + 10\right)\cdot 53^{2} + \left(49 a^{3} + 52 a^{2} + 34 a + 4\right)\cdot 53^{3} + \left(11 a^{3} + 12 a^{2} + 18 a + 49\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 41 a^{3} + 31 a^{2} + 14 a + 50 + \left(25 a^{3} + 20 a + 9\right)\cdot 53 + \left(46 a^{3} + 31 a^{2} + 10 a + 37\right)\cdot 53^{2} + \left(49 a^{3} + 52 a^{2} + 34 a + 39\right)\cdot 53^{3} + \left(11 a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 41 a^{3} + 31 a^{2} + 14 a + 16 + \left(25 a^{3} + 20 a + 24\right)\cdot 53 + \left(46 a^{3} + 31 a^{2} + 10 a + 52\right)\cdot 53^{2} + \left(49 a^{3} + 52 a^{2} + 34 a + 27\right)\cdot 53^{3} + \left(11 a^{3} + 12 a^{2} + 18 a + 32\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 27 a^{3} + 22 a^{2} + 45 a + 1 + \left(18 a^{3} + 25 a^{2} + 24 a + 15\right)\cdot 53 + \left(13 a^{3} + 25 a^{2} + 12 a + 48\right)\cdot 53^{2} + \left(23 a + 24\right)\cdot 53^{3} + \left(51 a^{3} + 18 a^{2} + 5 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 27 a^{3} + 22 a^{2} + 45 a + 8 + \left(18 a^{3} + 25 a^{2} + 24 a + 47\right)\cdot 53 + \left(13 a^{3} + 25 a^{2} + 12 a + 21\right)\cdot 53^{2} + \left(23 a + 7\right)\cdot 53^{3} + \left(51 a^{3} + 18 a^{2} + 5 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 27 a^{3} + 22 a^{2} + 45 a + 27 + \left(18 a^{3} + 25 a^{2} + 24 a + 8\right)\cdot 53 + \left(13 a^{3} + 25 a^{2} + 12 a + 37\right)\cdot 53^{2} + \left(23 a + 48\right)\cdot 53^{3} + \left(51 a^{3} + 18 a^{2} + 5 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 12 a^{3} + 22 a^{2} + 39 a + 24 + \left(27 a^{3} + 52 a^{2} + 32 a + 40\right)\cdot 53 + \left(6 a^{3} + 21 a^{2} + 42 a + 49\right)\cdot 53^{2} + \left(3 a^{3} + 18 a + 26\right)\cdot 53^{3} + \left(41 a^{3} + 40 a^{2} + 34 a + 46\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 12 a^{3} + 22 a^{2} + 39 a + 31 + \left(27 a^{3} + 52 a^{2} + 32 a + 19\right)\cdot 53 + \left(6 a^{3} + 21 a^{2} + 42 a + 23\right)\cdot 53^{2} + \left(3 a^{3} + 18 a + 9\right)\cdot 53^{3} + \left(41 a^{3} + 40 a^{2} + 34 a + 52\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 12 a^{3} + 22 a^{2} + 39 a + 50 + \left(27 a^{3} + 52 a^{2} + 32 a + 33\right)\cdot 53 + \left(6 a^{3} + 21 a^{2} + 42 a + 38\right)\cdot 53^{2} + \left(3 a^{3} + 18 a + 50\right)\cdot 53^{3} + \left(41 a^{3} + 40 a^{2} + 34 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 26 a^{3} + 31 a^{2} + 8 a + 13 + \left(34 a^{3} + 27 a^{2} + 28 a + 3\right)\cdot 53 + \left(39 a^{3} + 27 a^{2} + 40 a + 12\right)\cdot 53^{2} + \left(52 a^{3} + 52 a^{2} + 29 a + 6\right)\cdot 53^{3} + \left(a^{3} + 34 a^{2} + 47 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 26 a^{3} + 31 a^{2} + 8 a + 39 + \left(34 a^{3} + 27 a^{2} + 28 a + 49\right)\cdot 53 + \left(39 a^{3} + 27 a^{2} + 40 a\right)\cdot 53^{2} + \left(52 a^{3} + 52 a^{2} + 29 a + 30\right)\cdot 53^{3} + \left(a^{3} + 34 a^{2} + 47 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 26 a^{3} + 31 a^{2} + 8 a + 20 + \left(34 a^{3} + 27 a^{2} + 28 a + 35\right)\cdot 53 + \left(39 a^{3} + 27 a^{2} + 40 a + 38\right)\cdot 53^{2} + \left(52 a^{3} + 52 a^{2} + 29 a + 41\right)\cdot 53^{3} + \left(a^{3} + 34 a^{2} + 47 a + 2\right)\cdot 53^{4} +O\left(53^{ 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 values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,12)(6,11)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)(10,11,12)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)(10,12,11)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $4$ | $(1,10,7,4)(2,12,8,5)(3,11,9,6)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,4,7,10)(2,5,8,12)(3,6,9,11)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,8,3,7,2,9)(4,12,6,10,5,11)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,9,2,7,3,8)(4,11,5,10,6,12)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,6,8,10,3,5,7,11,2,4,9,12)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,5,9,10,2,6,7,12,3,4,8,11)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,11,8,4,3,12,7,6,2,10,9,5)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,12,9,4,2,11,7,5,3,10,8,6)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |