Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $216= 2^{3} \cdot 3^{3} $ |
| Artin number field: | Splitting field of $f= x^{12} - 2 x^{11} + x^{10} - 2 x^{9} + 3 x^{8} + 2 x^{7} - 6 x^{5} + x^{4} + 4 x^{3} - 2 x^{2} + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.2e3_3e2.6t1.3c1 |
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^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 3 a^{5} + 11 a^{4} + 4 a^{3} + 8 a^{2} + 8 a + 6 + \left(6 a^{5} + 3 a^{4} + 12 a^{3} + 8 a^{2} + 3 a + 5\right)\cdot 13 + \left(4 a^{5} + 8 a^{4} + 8 a^{3} + 7 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(2 a^{4} + 3 a^{3} + 6 a^{2} + 5 a + 2\right)\cdot 13^{3} + \left(3 a^{5} + 3 a^{4} + a^{3} + 11 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(3 a^{4} + 7 a^{3} + 3 a^{2} + 12 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 7 a^{5} + 2 a^{4} + 7 a^{3} + 10 a^{2} + 9 a + 9 + \left(6 a^{5} + 12 a^{4} + 12 a^{3} + 6 a^{2} + 12 a + 9\right)\cdot 13 + \left(a^{5} + 7 a^{4} + 6 a^{2} + a + 9\right)\cdot 13^{2} + \left(7 a^{5} + 2 a^{4} + 8 a^{3} + 8 a^{2} + 6 a + 8\right)\cdot 13^{3} + \left(a^{5} + a^{4} + 11 a^{3} + 10 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(4 a^{5} + 7 a^{3} + 11 a^{2} + 9 a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 8 a^{5} + 12 a^{4} + 6 a^{3} + 6 a^{2} + a + 2 + \left(9 a^{5} + 12 a^{4} + 12 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 13 + \left(3 a^{5} + a^{4} + 4 a^{3} + 12 a^{2} + 4 a + 2\right)\cdot 13^{2} + \left(12 a^{5} + 3 a^{4} + 3 a^{3} + 9 a^{2} + 7 a + 4\right)\cdot 13^{3} + \left(10 a^{5} + 7 a^{4} + a^{3} + 8 a + 6\right)\cdot 13^{4} + \left(3 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 11 a^{5} + 7 a^{3} + 12 a^{2} + 7 + \left(3 a^{5} + 9 a^{4} + 7 a^{3} + 2 a^{2} + 3 a + 9\right)\cdot 13 + \left(4 a^{5} + 8 a^{4} + 8 a^{3} + 2 a^{2} + 2 a\right)\cdot 13^{2} + \left(9 a^{5} + 3 a^{4} + a^{3} + a^{2} + 12 a + 7\right)\cdot 13^{3} + \left(9 a^{5} + 9 a^{3} + 5 a^{2} + 12 a + 8\right)\cdot 13^{4} + \left(5 a^{5} + 10 a^{4} + 10 a^{3} + 10 a^{2} + a + 7\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 9 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + \left(5 a^{5} + 9 a^{3} + 8 a^{2} + 2 a\right)\cdot 13 + \left(11 a^{5} + a^{4} + 7 a^{3} + 8 a^{2} + 6 a + 8\right)\cdot 13^{2} + \left(3 a^{5} + 7 a^{4} + 10 a^{3} + 5 a + 11\right)\cdot 13^{3} + \left(10 a^{5} + 6 a^{4} + 10 a^{3} + 6 a^{2} + 2 a + 11\right)\cdot 13^{4} + \left(4 a^{5} + 8 a^{4} + 10 a^{3} + 7 a + 7\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 6 a^{3} + 3 a^{2} + 7 a + 3 + \left(5 a^{5} + 4 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 11\right)\cdot 13 + \left(11 a^{4} + 9 a^{3} + 2 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(a^{5} + 10 a^{4} + 3 a^{3} + 10 a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(8 a^{5} + 6 a^{3} + 3 a^{2} + 5 a + 9\right)\cdot 13^{4} + \left(7 a^{5} + 6 a^{4} + 12 a^{2} + a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 7 a^{5} + 8 a^{3} + 10 a^{2} + 8 + \left(2 a^{5} + a^{4} + 8 a^{3} + 10 a^{2} + 9 a + 1\right)\cdot 13 + \left(4 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + 6\right)\cdot 13^{2} + \left(2 a^{5} + 4 a^{4} + 12 a^{3} + 10 a^{2} + a + 6\right)\cdot 13^{3} + \left(9 a^{5} + 5 a^{4} + 5 a^{3} + 8 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(7 a^{5} + a^{4} + 2 a^{2} + 2 a + 12\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ a^{5} + a^{4} + 9 a^{3} + 9 a^{2} + \left(12 a^{5} + 10 a^{4} + 8 a^{3} + 5 a^{2} + 6 a\right)\cdot 13 + \left(7 a^{5} + 12 a^{4} + 12 a^{3} + 4 a^{2} + a + 11\right)\cdot 13^{2} + \left(5 a^{5} + 3 a^{3} + 4 a^{2} + 5 a + 5\right)\cdot 13^{3} + \left(9 a^{5} + 4 a^{4} + 3 a^{2} + 5 a + 3\right)\cdot 13^{4} + \left(12 a^{5} + 3 a^{4} + 4 a^{3} + 5 a^{2} + 7 a + 10\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 5 a^{3} + 9 a^{2} + 8 a + 9 + \left(2 a^{5} + 12 a^{4} + 10 a^{3} + 2 a^{2} + 11 a + 1\right)\cdot 13 + \left(4 a^{5} + 12 a^{4} + 3 a^{3} + a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(11 a^{5} + 11 a^{4} + 7 a^{3} + 12 a^{2} + 3\right)\cdot 13^{3} + \left(4 a^{5} + 4 a^{4} + 5 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 13^{4} + \left(a^{4} + 9 a^{3} + 11 a^{2} + 10 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ a^{5} + 8 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 2 + \left(7 a^{5} + 6 a^{4} + 10 a^{3} + 7 a^{2} + a + 3\right)\cdot 13 + \left(12 a^{5} + a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 10\right)\cdot 13^{2} + \left(a^{5} + 10 a^{4} + 10 a^{3} + 12 a^{2} + 7 a + 11\right)\cdot 13^{3} + \left(10 a^{5} + 4 a^{4} + 8 a^{3} + 10 a + 2\right)\cdot 13^{4} + \left(3 a^{5} + 7 a^{4} + 7 a^{3} + 11 a^{2} + a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 11 a^{5} + 5 a^{4} + 11 a^{3} + 12 a^{2} + 3 a + 3 + \left(7 a^{4} + 2 a^{3} + 5 a^{2} + 6 a + 5\right)\cdot 13 + \left(10 a^{5} + 4 a^{4} + 11 a^{3} + 8 a^{2} + 6 a + 8\right)\cdot 13^{2} + \left(8 a^{5} + 6 a^{4} + 10 a^{2} + 4 a + 4\right)\cdot 13^{3} + \left(6 a^{5} + 3 a^{4} + 9 a^{3} + 5 a^{2} + 5 a + 10\right)\cdot 13^{4} + \left(6 a^{5} + 5 a^{4} + 6 a^{3} + 9 a^{2} + 12 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 7 a^{5} + 4 a^{4} + 2 a^{3} + 2 a^{2} + 9 a + 5 + \left(3 a^{5} + 11 a^{4} + a^{3} + 7 a^{2} + 4 a\right)\cdot 13 + \left(11 a^{4} + 2 a^{3} + 8 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(a^{5} + 11 a^{3} + 3 a^{2} + 11 a + 5\right)\cdot 13^{3} + \left(7 a^{5} + 10 a^{4} + 7 a^{3} + 4 a^{2} + a + 6\right)\cdot 13^{4} + \left(10 a^{5} + a^{4} + 2 a^{3} + 8 a^{2} + a\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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,10)(2,11)(3,12)(4,7)(5,8)(6,9)$ | $-2$ |
| $3$ | $2$ | $(1,11)(2,10)(3,5)(4,9)(6,7)(8,12)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,8)(4,6)(5,12)(7,9)(10,11)$ | $0$ |
| $1$ | $3$ | $(1,9,8)(2,7,3)(4,12,11)(5,10,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,8,9)(2,3,7)(4,11,12)(5,6,10)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,3,7)(4,11,12)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,7,3)(4,12,11)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,8,9)(2,7,3)(4,12,11)(5,6,10)$ | $-1$ |
| $1$ | $6$ | $(1,6,8,10,9,5)(2,4,3,11,7,12)$ | $2 \zeta_{3} + 2$ |
| $1$ | $6$ | $(1,5,9,10,8,6)(2,12,7,11,3,4)$ | $-2 \zeta_{3}$ |
| $2$ | $6$ | $(1,10)(2,12,7,11,3,4)(5,8)(6,9)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,10)(2,4,3,11,7,12)(5,8)(6,9)$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,5,9,10,8,6)(2,4,3,11,7,12)$ | $1$ |
| $3$ | $6$ | $(1,4,8,11,9,12)(2,6,3,10,7,5)$ | $0$ |
| $3$ | $6$ | $(1,12,9,11,8,4)(2,5,7,10,3,6)$ | $0$ |
| $3$ | $6$ | $(1,7,8,2,9,3)(4,5,11,6,12,10)$ | $0$ |
| $3$ | $6$ | $(1,3,9,2,8,7)(4,10,12,6,11,5)$ | $0$ |