Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | $140= 2^{2} \cdot 5 \cdot 7 $ |
| Artin number field: | Splitting field of $f= x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 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 $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{4} + 3 x^{2} + 12 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 9 a^{3} + 12 a^{2} + 5 a + 1 + \left(9 a^{2} + 6 a + 2\right)\cdot 13 + \left(5 a + 12\right)\cdot 13^{2} + \left(9 a^{3} + 7 a^{2} + 8 a + 12\right)\cdot 13^{3} + \left(3 a^{3} + 2 a^{2} + 8 a + 12\right)\cdot 13^{4} + \left(a^{3} + a^{2} + a + 5\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 11 a^{2} + 12 a + \left(9 a^{3} + 2 a^{2} + 11 a + 3\right)\cdot 13 + \left(2 a^{3} + 5 a^{2} + 8 a + 11\right)\cdot 13^{2} + \left(7 a^{3} + 6 a^{2} + 8 a + 1\right)\cdot 13^{3} + \left(2 a^{3} + 12 a^{2} + 8 a + 1\right)\cdot 13^{4} + \left(8 a^{3} + a^{2} + 5 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 9 a^{3} + 12 a^{2} + 5 a + 11 + \left(9 a^{2} + 6 a + 3\right)\cdot 13 + 5 a\cdot 13^{2} + \left(9 a^{3} + 7 a^{2} + 8 a + 6\right)\cdot 13^{3} + \left(3 a^{3} + 2 a^{2} + 8 a + 9\right)\cdot 13^{4} + \left(a^{3} + a^{2} + a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 11 a^{2} + 12 a + 3 + \left(9 a^{3} + 2 a^{2} + 11 a + 1\right)\cdot 13 + \left(2 a^{3} + 5 a^{2} + 8 a + 10\right)\cdot 13^{2} + \left(7 a^{3} + 6 a^{2} + 8 a + 8\right)\cdot 13^{3} + \left(2 a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 13^{4} + \left(8 a^{3} + a^{2} + 5 a + 4\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 4 a^{3} + a^{2} + 8 a + 11 + \left(12 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 13 + \left(12 a^{3} + 12 a^{2} + 7 a + 8\right)\cdot 13^{2} + \left(3 a^{3} + 5 a^{2} + 4 a + 11\right)\cdot 13^{3} + \left(9 a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 11 a + 5\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 2 a^{2} + a + 8 + \left(4 a^{3} + 10 a^{2} + a + 9\right)\cdot 13 + \left(10 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 13^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 11\right)\cdot 13^{3} + \left(10 a^{3} + 4 a + 9\right)\cdot 13^{4} + \left(4 a^{3} + 11 a^{2} + 7 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 4 a^{3} + a^{2} + 8 a + 10 + \left(12 a^{3} + 3 a^{2} + 6 a + 8\right)\cdot 13 + \left(12 a^{3} + 12 a^{2} + 7 a + 8\right)\cdot 13^{2} + \left(3 a^{3} + 5 a^{2} + 4 a + 7\right)\cdot 13^{3} + \left(9 a^{3} + 10 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 11 a + 10\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 2 a^{2} + a + 9 + \left(4 a^{3} + 10 a^{2} + a + 12\right)\cdot 13 + \left(10 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 13^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 2\right)\cdot 13^{3} + \left(10 a^{3} + 4 a + 12\right)\cdot 13^{4} + \left(4 a^{3} + 11 a^{2} + 7 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 2 a^{2} + a + 6 + \left(4 a^{3} + 10 a^{2} + a + 1\right)\cdot 13 + \left(10 a^{3} + 7 a^{2} + 4 a + 12\right)\cdot 13^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 8\right)\cdot 13^{3} + \left(10 a^{3} + 4 a + 8\right)\cdot 13^{4} + \left(4 a^{3} + 11 a^{2} + 7 a + 4\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 11 a^{2} + 12 a + 2 + \left(9 a^{3} + 2 a^{2} + 11 a + 11\right)\cdot 13 + \left(2 a^{3} + 5 a^{2} + 8 a + 9\right)\cdot 13^{2} + \left(7 a^{3} + 6 a^{2} + 8 a + 4\right)\cdot 13^{3} + \left(2 a^{3} + 12 a^{2} + 8 a + 2\right)\cdot 13^{4} + \left(8 a^{3} + a^{2} + 5 a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 4 a^{3} + a^{2} + 8 a + 8 + \left(12 a^{3} + 3 a^{2} + 6 a\right)\cdot 13 + \left(12 a^{3} + 12 a^{2} + 7 a + 10\right)\cdot 13^{2} + \left(3 a^{3} + 5 a^{2} + 4 a + 4\right)\cdot 13^{3} + \left(9 a^{3} + 10 a^{2} + 4 a\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 11 a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 9 a^{3} + 12 a^{2} + 5 a + \left(9 a^{2} + 6 a + 12\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(9 a^{3} + 7 a^{2} + 8 a + 8\right)\cdot 13^{3} + \left(3 a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 13^{4} + \left(a^{3} + a^{2} + a + 10\right)\cdot 13^{5} +O\left(13^{ 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 | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,5)(2,9)(3,11)(4,8)(6,10)(7,12)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,12,3)(2,4,10)(5,7,11)(6,9,8)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
| $1$ | $3$ | $(1,3,12)(2,10,4)(5,11,7)(6,8,9)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $4$ | $(1,8,5,4)(2,3,9,11)(6,7,10,12)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,4,5,8)(2,11,9,3)(6,12,10,7)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,11,12,5,3,7)(2,6,4,9,10,8)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,7,3,5,12,11)(2,8,10,9,4,6)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,10,11,8,12,2,5,6,3,4,7,9)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $1$ | $12$ | $(1,2,7,8,3,10,5,9,12,4,11,6)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,6,11,4,12,9,5,10,3,8,7,2)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,9,7,4,3,6,5,2,12,8,11,10)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |