Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $180= 2^{2} \cdot 3^{2} \cdot 5 $ |
| Artin number field: | Splitting field of $f= x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1 $ 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 $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{4} + 7 x^{2} + 10 x + 3 $
Roots:
| $r_{ 1 }$ | $=$ | $ 10 a^{3} + 5 a^{2} + 10 a + 9 + \left(3 a^{3} + 16 a^{2} + 3 a + 11\right)\cdot 17 + \left(14 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 17^{2} + \left(5 a^{2} + 7 a + 16\right)\cdot 17^{3} + \left(7 a^{3} + 2 a^{2} + 2 a + 14\right)\cdot 17^{4} + \left(10 a^{3} + 15 a + 2\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 7 a^{3} + 12 a^{2} + 7 a + 5 + \left(13 a^{3} + 13 a + 10\right)\cdot 17 + \left(2 a^{3} + 9 a^{2} + 14 a + 6\right)\cdot 17^{2} + \left(16 a^{3} + 11 a^{2} + 9 a + 2\right)\cdot 17^{3} + \left(9 a^{3} + 14 a^{2} + 14 a + 14\right)\cdot 17^{4} + \left(6 a^{3} + 16 a^{2} + a + 1\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 5 a^{3} + 4 a^{2} + 11 a + 12 + \left(12 a^{3} + 13 a^{2} + 13 a + 14\right)\cdot 17 + \left(9 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 17^{2} + \left(14 a^{3} + 9 a^{2} + 4 a + 13\right)\cdot 17^{3} + \left(a^{3} + a^{2} + 6 a + 15\right)\cdot 17^{4} + \left(13 a^{3} + 9 a^{2} + 15 a + 14\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 12 a^{3} + 13 a^{2} + 6 a + 12 + \left(4 a^{3} + 3 a^{2} + 3 a + 6\right)\cdot 17 + \left(7 a^{3} + 7 a^{2} + 10 a + 8\right)\cdot 17^{2} + \left(2 a^{3} + 7 a^{2} + 12 a + 12\right)\cdot 17^{3} + \left(15 a^{3} + 15 a^{2} + 10 a + 4\right)\cdot 17^{4} + \left(3 a^{3} + 7 a^{2} + a + 1\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 5 a^{3} + 4 a^{2} + 11 a + 2 + \left(12 a^{3} + 13 a^{2} + 13 a + 15\right)\cdot 17 + \left(9 a^{3} + 9 a^{2} + 6 a + 10\right)\cdot 17^{2} + \left(14 a^{3} + 9 a^{2} + 4 a + 6\right)\cdot 17^{3} + \left(a^{3} + a^{2} + 6 a + 7\right)\cdot 17^{4} + \left(13 a^{3} + 9 a^{2} + 15 a + 3\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 10 a^{3} + 5 a^{2} + 10 a + 2 + \left(3 a^{3} + 16 a^{2} + 3 a + 11\right)\cdot 17 + \left(14 a^{3} + 7 a^{2} + 2 a + 4\right)\cdot 17^{2} + \left(5 a^{2} + 7 a + 6\right)\cdot 17^{3} + \left(7 a^{3} + 2 a^{2} + 2 a + 6\right)\cdot 17^{4} + \left(10 a^{3} + 15 a + 14\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 12 a^{3} + 13 a^{2} + 6 a + 11 + \left(4 a^{3} + 3 a^{2} + 3 a + 9\right)\cdot 17 + \left(7 a^{3} + 7 a^{2} + 10 a + 9\right)\cdot 17^{2} + \left(2 a^{3} + 7 a^{2} + 12 a + 16\right)\cdot 17^{3} + \left(15 a^{3} + 15 a^{2} + 10 a + 10\right)\cdot 17^{4} + \left(3 a^{3} + 7 a^{2} + a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 7 a^{3} + 12 a^{2} + 7 a + 4 + \left(13 a^{3} + 13 a + 13\right)\cdot 17 + \left(2 a^{3} + 9 a^{2} + 14 a + 7\right)\cdot 17^{2} + \left(16 a^{3} + 11 a^{2} + 9 a + 6\right)\cdot 17^{3} + \left(9 a^{3} + 14 a^{2} + 14 a + 3\right)\cdot 17^{4} + \left(6 a^{3} + 16 a^{2} + a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 7 a^{3} + 12 a^{2} + 7 a + 11 + \left(13 a^{3} + 13 a + 13\right)\cdot 17 + \left(2 a^{3} + 9 a^{2} + 14 a + 15\right)\cdot 17^{2} + \left(16 a^{3} + 11 a^{2} + 9 a + 16\right)\cdot 17^{3} + \left(9 a^{3} + 14 a^{2} + 14 a + 11\right)\cdot 17^{4} + \left(6 a^{3} + 16 a^{2} + a + 15\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 12 a^{3} + 13 a^{2} + 6 a + 1 + \left(4 a^{3} + 3 a^{2} + 3 a + 10\right)\cdot 17 + \left(7 a^{3} + 7 a^{2} + 10 a\right)\cdot 17^{2} + \left(2 a^{3} + 7 a^{2} + 12 a + 10\right)\cdot 17^{3} + \left(15 a^{3} + 15 a^{2} + 10 a + 2\right)\cdot 17^{4} + \left(3 a^{3} + 7 a^{2} + a + 15\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 5 a^{3} + 4 a^{2} + 11 a + 13 + \left(12 a^{3} + 13 a^{2} + 13 a + 11\right)\cdot 17 + \left(9 a^{3} + 9 a^{2} + 6 a + 1\right)\cdot 17^{2} + \left(14 a^{3} + 9 a^{2} + 4 a + 9\right)\cdot 17^{3} + \left(a^{3} + a^{2} + 6 a + 9\right)\cdot 17^{4} + \left(13 a^{3} + 9 a^{2} + 15 a + 6\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 10 a^{3} + 5 a^{2} + 10 a + 3 + \left(3 a^{3} + 16 a^{2} + 3 a + 8\right)\cdot 17 + \left(14 a^{3} + 7 a^{2} + 2 a + 3\right)\cdot 17^{2} + \left(5 a^{2} + 7 a + 2\right)\cdot 17^{3} + \left(7 a^{3} + 2 a^{2} + 2 a\right)\cdot 17^{4} + \left(10 a^{3} + 15 a + 6\right)\cdot 17^{5} +O\left(17^{ 6 }\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 | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,9)(2,12)(3,7)(4,11)(5,10)(6,8)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,12,6)(2,8,9)(3,5,11)(4,7,10)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $3$ | $(1,6,12)(2,9,8)(3,11,5)(4,10,7)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
| $1$ | $4$ | $(1,5,9,10)(2,4,12,11)(3,8,7,6)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,10,9,5)(2,11,12,4)(3,6,7,8)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,2,6,9,12,8)(3,10,11,7,5,4)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,8,12,9,6,2)(3,4,5,7,11,10)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,7,2,5,6,4,9,3,12,10,8,11)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,4,8,5,12,7,9,11,6,10,2,3)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,3,2,10,6,11,9,7,12,5,8,4)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,11,8,10,12,3,9,4,6,5,2,7)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |