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} + 2 x^{10} + 71 x^{8} - 60 x^{7} - 166 x^{6} - 468 x^{5} + 234 x^{4} + 2196 x^{3} + 5160 x^{2} + 3408 x + 2417 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $C_{12}$ |
| Parity: | Odd |
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:
| $r_{ 1 }$ | $=$ | $ 42 a^{3} + 24 a^{2} + 4 a + 13 + \left(7 a^{3} + 27 a^{2} + 9 a + 30\right)\cdot 53 + \left(43 a^{3} + 37 a^{2} + 16 a + 39\right)\cdot 53^{2} + \left(30 a^{3} + 33 a^{2} + 38 a + 35\right)\cdot 53^{3} + \left(13 a^{3} + 22 a^{2} + 10 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 42 a^{3} + 24 a^{2} + 4 a + 40 + \left(7 a^{3} + 27 a^{2} + 9 a + 36\right)\cdot 53 + \left(43 a^{3} + 37 a^{2} + 16 a + 50\right)\cdot 53^{2} + \left(30 a^{3} + 33 a^{2} + 38 a + 11\right)\cdot 53^{3} + \left(13 a^{3} + 22 a^{2} + 10 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 42 a^{3} + 24 a^{2} + 4 a + 47 + \left(7 a^{3} + 27 a^{2} + 9 a + 15\right)\cdot 53 + \left(43 a^{3} + 37 a^{2} + 16 a + 24\right)\cdot 53^{2} + \left(30 a^{3} + 33 a^{2} + 38 a + 47\right)\cdot 53^{3} + \left(13 a^{3} + 22 a^{2} + 10 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 38 a^{3} + 29 a^{2} + 28 a + 28 + \left(32 a^{3} + 40 a^{2} + 22 a + 37\right)\cdot 53 + \left(44 a^{3} + 11 a^{2} + 7 a + 29\right)\cdot 53^{2} + \left(44 a^{3} + 4 a^{2} + 6 a + 40\right)\cdot 53^{3} + \left(41 a^{3} + 22 a^{2} + 33 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 38 a^{3} + 29 a^{2} + 28 a + 1 + \left(32 a^{3} + 40 a^{2} + 22 a + 31\right)\cdot 53 + \left(44 a^{3} + 11 a^{2} + 7 a + 18\right)\cdot 53^{2} + \left(44 a^{3} + 4 a^{2} + 6 a + 11\right)\cdot 53^{3} + \left(41 a^{3} + 22 a^{2} + 33 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 38 a^{3} + 29 a^{2} + 28 a + 35 + \left(32 a^{3} + 40 a^{2} + 22 a + 16\right)\cdot 53 + \left(44 a^{3} + 11 a^{2} + 7 a + 3\right)\cdot 53^{2} + \left(44 a^{3} + 4 a^{2} + 6 a + 23\right)\cdot 53^{3} + \left(41 a^{3} + 22 a^{2} + 33 a + 20\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 11 a^{3} + 29 a^{2} + 49 a + 27 + \left(45 a^{3} + 25 a^{2} + 43 a + 34\right)\cdot 53 + \left(9 a^{3} + 15 a^{2} + 36 a + 9\right)\cdot 53^{2} + \left(22 a^{3} + 19 a^{2} + 14 a + 19\right)\cdot 53^{3} + \left(39 a^{3} + 30 a^{2} + 42 a + 9\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 11 a^{3} + 29 a^{2} + 49 a + \left(45 a^{3} + 25 a^{2} + 43 a + 28\right)\cdot 53 + \left(9 a^{3} + 15 a^{2} + 36 a + 51\right)\cdot 53^{2} + \left(22 a^{3} + 19 a^{2} + 14 a + 42\right)\cdot 53^{3} + \left(39 a^{3} + 30 a^{2} + 42 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 11 a^{3} + 29 a^{2} + 49 a + 34 + \left(45 a^{3} + 25 a^{2} + 43 a + 13\right)\cdot 53 + \left(9 a^{3} + 15 a^{2} + 36 a + 36\right)\cdot 53^{2} + \left(22 a^{3} + 19 a^{2} + 14 a + 1\right)\cdot 53^{3} + \left(39 a^{3} + 30 a^{2} + 42 a + 15\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 15 a^{3} + 24 a^{2} + 25 a + 46 + \left(20 a^{3} + 12 a^{2} + 30 a + 12\right)\cdot 53 + \left(8 a^{3} + 41 a^{2} + 45 a + 4\right)\cdot 53^{2} + \left(8 a^{3} + 48 a^{2} + 46 a + 26\right)\cdot 53^{3} + \left(11 a^{3} + 30 a^{2} + 19 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 15 a^{3} + 24 a^{2} + 25 a + 12 + \left(20 a^{3} + 12 a^{2} + 30 a + 27\right)\cdot 53 + \left(8 a^{3} + 41 a^{2} + 45 a + 19\right)\cdot 53^{2} + \left(8 a^{3} + 48 a^{2} + 46 a + 14\right)\cdot 53^{3} + \left(11 a^{3} + 30 a^{2} + 19 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 15 a^{3} + 24 a^{2} + 25 a + 39 + \left(20 a^{3} + 12 a^{2} + 30 a + 33\right)\cdot 53 + \left(8 a^{3} + 41 a^{2} + 45 a + 30\right)\cdot 53^{2} + \left(8 a^{3} + 48 a^{2} + 46 a + 43\right)\cdot 53^{3} + \left(11 a^{3} + 30 a^{2} + 19 a + 27\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 | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,9)(4,12)(5,11)(6,10)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)(10,11,12)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)(7,8,9)(10,12,11)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $4$ | $(1,5,8,11)(2,4,7,12)(3,6,9,10)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,11,8,5)(2,12,7,4)(3,10,9,6)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,9,2,8,3,7)(4,11,6,12,5,10)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,7,3,8,2,9)(4,10,5,12,6,11)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,12,9,5,2,10,8,4,3,11,7,6)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,10,7,5,3,12,8,6,2,11,9,4)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,4,9,11,2,6,8,12,3,5,7,10)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,6,7,11,3,4,8,10,2,5,9,12)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |