Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $665= 5 \cdot 7 \cdot 19 $ |
| Artin number field: | Splitting field of $f= x^{12} - x^{11} + 45 x^{10} - 25 x^{9} + 1941 x^{8} - 4660 x^{7} + 88464 x^{6} - 169280 x^{5} + 3763456 x^{4} - 5550080 x^{3} + 8192000 x^{2} - 11534336 x + 16777216 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $C_{12}$ |
| Parity: | Odd |
| Corresponding Dirichlet character: | \(\chi_{665}(653,\cdot)\) |
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:
| $r_{ 1 }$ | $=$ | $ 8 a^{3} + 37 a^{2} + 5 a + 2 + \left(19 a^{3} + 47 a^{2} + 12 a + 52\right)\cdot 53 + \left(37 a^{3} + 36 a^{2} + 22 a + 29\right)\cdot 53^{2} + \left(14 a^{3} + 40 a^{2} + 24 a + 37\right)\cdot 53^{3} + \left(44 a^{3} + 10 a^{2} + 31 a + 36\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 48 a^{3} + 7 a^{2} + 34 a + 8 + \left(36 a^{3} + 11 a^{2} + 14 a + 19\right)\cdot 53 + \left(8 a^{3} + 44 a^{2} + 37 a + 17\right)\cdot 53^{2} + \left(37 a^{3} + 25 a^{2} + 26 a + 36\right)\cdot 53^{3} + \left(41 a^{3} + 33 a^{2} + 24 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 17 a^{3} + 8 a^{2} + 34 a + 22 + \left(44 a^{2} + 26 a + 29\right)\cdot 53 + \left(6 a^{3} + 4 a^{2} + 35 a + 25\right)\cdot 53^{2} + \left(21 a^{3} + 49 a^{2} + 21 a + 19\right)\cdot 53^{3} + \left(40 a^{3} + 41 a^{2} + 26 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 46 a^{3} + 35 a^{2} + 13 a + 24 + \left(a^{3} + 46 a^{2} + 51 a + 28\right)\cdot 53 + \left(35 a^{3} + 50 a^{2} + 43 a + 29\right)\cdot 53^{2} + \left(18 a^{3} + 9 a^{2} + 41 a + 20\right)\cdot 53^{3} + \left(47 a^{3} + 20 a^{2} + 11 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 14 a^{3} + 25 a^{2} + 22 a + 30 + \left(4 a^{3} + 9 a^{2} + 24 a + 7\right)\cdot 53 + \left(21 a^{3} + 34 a^{2} + 37 a + 13\right)\cdot 53^{2} + \left(46 a^{3} + 33 a^{2} + 19 a + 37\right)\cdot 53^{3} + \left(8 a^{3} + 48 a^{2} + 28 a + 42\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 43 a^{3} + 14 a^{2} + 33 a + 12 + \left(32 a^{3} + 8 a^{2} + 31 a + 47\right)\cdot 53 + \left(23 a^{3} + 20 a^{2} + 49 a + 48\right)\cdot 53^{2} + \left(14 a^{3} + 20 a^{2} + 5 a + 20\right)\cdot 53^{3} + \left(16 a^{3} + 33 a^{2} + 16 a + 52\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 47 a^{3} + 19 a^{2} + 9 a + 52 + \left(25 a^{2} + 3 a + 13\right)\cdot 53 + \left(40 a^{3} + 11 a^{2} + 9 a + 44\right)\cdot 53^{2} + \left(40 a^{3} + 37 a^{2} + 39 a + 48\right)\cdot 53^{3} + \left(43 a^{3} + 52 a^{2} + 14 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 42 a^{3} + 22 a^{2} + 13 a + 37 + \left(17 a^{3} + 18 a^{2} + 43 a + 10\right)\cdot 53 + \left(26 a^{3} + 31 a^{2} + 48 a + 38\right)\cdot 53^{2} + \left(30 a^{3} + 36 a^{2} + 33 a\right)\cdot 53^{3} + \left(23 a^{3} + 10 a^{2} + 34 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 34 a^{3} + 42 a^{2} + 5 a + 50 + \left(32 a^{3} + 41 a^{2} + 11 a + 38\right)\cdot 53 + \left(22 a^{3} + 52 a^{2} + 39 a + 25\right)\cdot 53^{2} + \left(29 a^{3} + 31 a^{2} + 20 a + 20\right)\cdot 53^{3} + \left(30 a^{3} + 33 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 23 a^{3} + 42 a^{2} + 46 a + 36 + \left(49 a^{3} + 29 a^{2} + 49 a + 9\right)\cdot 53 + \left(35 a^{3} + 32 a^{2} + 28 a + 52\right)\cdot 53^{2} + \left(19 a^{3} + 33 a^{2} + 3 a + 49\right)\cdot 53^{3} + \left(46 a^{3} + 41 a^{2} + 35 a + 44\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 16 a^{3} + 20 a^{2} + 29 a + 38 + \left(6 a^{2} + 40 a + 39\right)\cdot 53 + \left(33 a^{3} + 8 a^{2} + 2 a + 23\right)\cdot 53^{2} + \left(17 a^{3} + 17 a^{2} + 34 a + 17\right)\cdot 53^{3} + \left(37 a^{3} + 30 a\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 33 a^{3} + 47 a^{2} + 22 a + 8 + \left(15 a^{3} + 28 a^{2} + 9 a + 21\right)\cdot 53 + \left(28 a^{3} + 43 a^{2} + 16 a + 22\right)\cdot 53^{2} + \left(27 a^{3} + 34 a^{2} + 46 a + 8\right)\cdot 53^{3} + \left(43 a^{3} + 23 a^{2} + 30 a + 43\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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-1$ |
| $1$ | $3$ | $(1,8,5)(2,11,7)(3,10,6)(4,12,9)$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $3$ | $(1,5,8)(2,7,11)(3,6,10)(4,9,12)$ | $-\zeta_{12}^{2}$ |
| $1$ | $4$ | $(1,9,7,3)(2,10,8,4)(5,12,11,6)$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,3,7,9)(2,4,8,10)(5,6,11,12)$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,11,8,7,5,2)(3,12,10,9,6,4)$ | $\zeta_{12}^{2}$ |
| $1$ | $6$ | $(1,2,5,7,8,11)(3,4,6,9,10,12)$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $12$ | $(1,10,11,9,8,6,7,4,5,3,2,12)$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,6,2,9,5,10,7,12,8,3,11,4)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,4,11,3,8,12,7,10,5,9,2,6)$ | $\zeta_{12}$ |
| $1$ | $12$ | $(1,12,2,3,5,4,7,6,8,9,11,10)$ | $\zeta_{12}^{3} - \zeta_{12}$ |