Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $155= 5 \cdot 31 $ |
| Artin number field: | Splitting field of $f= x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $C_{12}$ |
| Parity: | Odd |
| Corresponding Dirichlet character: | \(\chi_{155}(98,\cdot)\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{4} + 3 x^{2} + 19 x + 5 $
Roots:
| $r_{ 1 }$ | $=$ | $ 17 a^{3} + 16 a^{2} + 19 a + 22 + \left(14 a^{3} + 4 a^{2} + 17 a + 16\right)\cdot 23 + \left(2 a^{3} + 18 a^{2} + 15 a + 8\right)\cdot 23^{2} + \left(16 a^{3} + 3 a^{2} + a + 16\right)\cdot 23^{3} + \left(a^{3} + 12 a^{2} + 22 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 6 a^{3} + 12 a^{2} + 4 a + 10 + \left(19 a^{3} + 6 a^{2} + 2 a + 20\right)\cdot 23 + \left(2 a^{3} + 16 a^{2} + 2 a + 9\right)\cdot 23^{2} + \left(21 a^{3} + 18 a^{2} + 21 a + 21\right)\cdot 23^{3} + \left(12 a^{3} + 11 a^{2} + 15 a + 18\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 17 a^{3} + 11 a^{2} + 18 a + 3 + \left(18 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 23 + \left(3 a^{3} + 3 a^{2} + 13 a + 20\right)\cdot 23^{2} + \left(20 a^{3} + 17 a^{2} + 11 a + 7\right)\cdot 23^{3} + \left(14 a^{3} + 4 a^{2} + a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 11 a^{3} + 13 a^{2} + 3 a + 8 + \left(2 a^{3} + 6 a^{2} + 22 a + 11\right)\cdot 23 + \left(a^{3} + 9 a^{2} + 16 a + 3\right)\cdot 23^{2} + \left(15 a^{3} + 22 a^{2} + 9 a + 15\right)\cdot 23^{3} + \left(11 a^{2} + 13 a + 22\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 5 a^{3} + 10 a^{2} + 11 a + 16 + \left(10 a^{3} + a^{2} + 13\right)\cdot 23 + \left(3 a^{3} + 11 a^{2} + 6 a + 21\right)\cdot 23^{2} + \left(10 a^{3} + 18 a^{2} + 13 a + 8\right)\cdot 23^{3} + \left(10 a^{3} + 15 a^{2} + 3 a + 22\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 3 a^{3} + 15 a^{2} + 2 a + 12 + \left(13 a^{3} + 19 a^{2} + 20 a + 4\right)\cdot 23 + \left(18 a^{3} + 9 a^{2} + 4 a\right)\cdot 23^{2} + \left(20 a^{3} + 8 a^{2} + 14 a + 13\right)\cdot 23^{3} + \left(4 a^{3} + 22 a^{2} + 10 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 13 a^{3} + 7 a^{2} + 14 a + 22 + \left(17 a^{3} + 20 a^{2} + 13 a + 2\right)\cdot 23 + \left(15 a^{3} + 19 a^{2} + 8 a + 9\right)\cdot 23^{2} + \left(a^{3} + 10 a^{2} + 2 a + 3\right)\cdot 23^{3} + \left(8 a^{3} + 4 a^{2} + 7 a + 6\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 11 a^{3} + 22 a^{2} + 15 a + 3 + \left(20 a^{3} + 12 a^{2} + 19 a + 19\right)\cdot 23 + \left(8 a^{3} + 11 a^{2} + 2 a + 18\right)\cdot 23^{2} + \left(4 a^{3} + 6 a^{2} + 6 a + 2\right)\cdot 23^{3} + \left(a^{3} + 21 a^{2} + 8 a + 8\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 3 a^{3} + 6 a^{2} + 14 a + 10 + \left(11 a^{3} + 13 a^{2} + a + 12\right)\cdot 23 + \left(10 a^{2} + 22 a + 19\right)\cdot 23^{2} + \left(12 a^{3} + 19 a^{2} + 7\right)\cdot 23^{3} + \left(4 a^{3} + 22 a^{2} + 6 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 14 a^{3} + a^{2} + 17 a + 10 + \left(17 a^{3} + 19 a^{2} + 7\right)\cdot 23 + \left(16 a^{3} + 10 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(a^{3} + 6 a^{2} + 16 a + 8\right)\cdot 23^{3} + \left(a^{3} + 10 a^{2} + 17 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ a^{3} + 20 a^{2} + 17 a + 7 + \left(15 a^{3} + 18 a^{2} + 19 a + 2\right)\cdot 23 + \left(7 a^{3} + 12 a^{2} + 13 a + 3\right)\cdot 23^{2} + \left(5 a^{3} + 18 a^{2} + 3 a + 17\right)\cdot 23^{3} + \left(5 a^{3} + 7 a^{2} + 14 a + 16\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 14 a^{3} + 5 a^{2} + 4 a + 16 + \left(5 a^{2} + 8 a + 14\right)\cdot 23 + \left(10 a^{3} + 4 a^{2} + 12 a + 1\right)\cdot 23^{2} + \left(9 a^{3} + 10 a^{2} + 13 a + 15\right)\cdot 23^{3} + \left(3 a^{3} + 15 a^{2} + 17 a + 13\right)\cdot 23^{4} +O\left(23^{ 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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,11)(2,9)(3,8)(4,6)(5,12)(7,10)$ | $-1$ |
| $1$ | $3$ | $(1,10,6)(2,8,5)(3,12,9)(4,11,7)$ | $-\zeta_{12}^{2}$ |
| $1$ | $3$ | $(1,6,10)(2,5,8)(3,9,12)(4,7,11)$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $4$ | $(1,8,11,3)(2,4,9,6)(5,7,12,10)$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,3,11,8)(2,6,9,4)(5,10,12,7)$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,4,10,11,6,7)(2,12,8,9,5,3)$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,7,6,11,10,4)(2,3,5,9,8,12)$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,5,4,3,10,2,11,12,6,8,7,9)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,2,7,3,6,5,11,9,10,8,4,12)$ | $\zeta_{12}$ |
| $1$ | $12$ | $(1,12,4,8,10,9,11,5,6,3,7,2)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,9,7,8,6,12,11,2,10,3,4,5)$ | $-\zeta_{12}$ |