Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $297= 3^{3} \cdot 11 $ |
| Artin number field: | Splitting field of $f= x^{12} + x^{10} - 4 x^{9} - 2 x^{8} - 3 x^{7} + 9 x^{6} + 4 x^{5} + 12 x^{4} - 7 x^{3} - 2 x^{2} - x + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.3e2_11.6t1.2c2 |
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} + 5 a^{3} + 9 a^{2} + 6 a + 7 + \left(9 a^{5} + 4 a^{4} + 2 a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 13 + \left(2 a^{5} + 5 a^{3} + 10 a^{2} + 12 a + 5\right)\cdot 13^{2} + \left(12 a^{5} + 2 a^{4} + 12 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(9 a^{5} + 5 a^{4} + 11 a^{3} + 5 a^{2} + 4 a\right)\cdot 13^{4} + \left(10 a^{5} + 11 a^{4} + 4 a^{3} + 12 a^{2} + 11 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 5 a^{4} + 2 a^{3} + 11 a^{2} + 6 a + 12 + \left(5 a^{4} + 11 a^{3} + 4 a^{2} + 5 a + 12\right)\cdot 13 + \left(11 a^{5} + 7 a^{4} + 9 a^{2} + a + 9\right)\cdot 13^{2} + \left(a^{4} + 11 a^{3} + 2 a^{2} + 4 a + 11\right)\cdot 13^{3} + \left(6 a^{5} + 11 a^{4} + 5 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 13^{4} + \left(9 a^{5} + 6 a^{4} + 9 a^{3} + 4 a^{2} + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 7 a^{5} + 4 a^{3} + 9 a^{2} + 6 a + 4 + \left(11 a^{5} + 2 a^{4} + 11 a^{3} + 7 a^{2} + 10 a + 12\right)\cdot 13 + \left(8 a^{5} + 10 a^{4} + 2 a^{3} + 6 a^{2} + 7 a + 9\right)\cdot 13^{2} + \left(4 a^{5} + 6 a^{4} + 10 a^{3} + 8 a^{2} + 3 a\right)\cdot 13^{3} + \left(7 a^{5} + 9 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(10 a^{5} + 5 a^{4} + 10 a^{3} + 4 a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 6 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + a + 10 + \left(a^{5} + 5 a^{4} + 3 a^{3} + 10 a\right)\cdot 13 + \left(6 a^{5} + 8 a^{4} + 9 a^{3} + 10 a^{2} + 3 a + 6\right)\cdot 13^{2} + \left(7 a^{5} + 4 a^{4} + 4 a^{3} + a^{2} + 5 a\right)\cdot 13^{3} + \left(12 a^{5} + 5 a^{4} + 11 a^{3} + 11 a^{2} + 2 a + 12\right)\cdot 13^{4} + \left(5 a^{5} + 5 a^{3} + 7 a^{2} + 8 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 2 a^{4} + 6 a^{3} + 7 a^{2} + 5 a + 10 + \left(6 a^{4} + 3 a^{3} + 5 a^{2} + 4 a + 1\right)\cdot 13 + \left(7 a^{5} + 7 a^{4} + 7 a^{3} + 4 a^{2} + 9 a + 10\right)\cdot 13^{2} + \left(6 a^{5} + 7 a^{4} + 10 a^{3} + 5 a^{2} + 10 a + 5\right)\cdot 13^{3} + \left(2 a^{5} + 9 a^{4} + 11 a^{3} + 6 a^{2} + 11\right)\cdot 13^{4} + \left(4 a^{5} + 6 a^{4} + a^{3} + 8 a^{2} + 12 a + 12\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 8 a^{5} + 12 a^{3} + a^{2} + 5 a + 12 + \left(7 a^{5} + 6 a^{4} + 12 a^{3} + 5 a^{2} + 6 a + 2\right)\cdot 13 + \left(2 a^{2} + 5 a + 7\right)\cdot 13^{2} + \left(10 a^{5} + a^{4} + 10 a^{3} + 4 a^{2} + 5 a\right)\cdot 13^{3} + \left(6 a^{5} + 12 a^{3} + 3 a^{2} + 8 a + 3\right)\cdot 13^{4} + \left(3 a^{5} + 8 a^{4} + 3 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 5 a^{5} + 11 a^{4} + 8 a^{3} + 5 a^{2} + 3 a + 4 + \left(5 a^{5} + 9 a^{3} + 2 a^{2} + 2 a + 8\right)\cdot 13 + \left(5 a^{5} + 5 a^{4} + 4 a^{3} + 6 a^{2} + 11 a + 8\right)\cdot 13^{2} + \left(9 a^{5} + 4 a^{4} + 5 a^{3} + 3 a^{2} + 9 a + 6\right)\cdot 13^{3} + \left(3 a^{5} + 3 a^{4} + a^{3} + 3 a^{2} + 3 a + 11\right)\cdot 13^{4} + \left(5 a^{5} + 11 a^{4} + 7 a^{3} + 2 a^{2} + 7 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 10 a^{5} + 4 a^{4} + a^{3} + 5 a^{2} + 2 a + 9 + \left(6 a^{5} + 2 a^{4} + 10 a^{3} + 3 a^{2} + 4 a + 3\right)\cdot 13 + \left(2 a^{5} + 10 a^{4} + 12 a^{3} + 6 a^{2} + 4 a + 1\right)\cdot 13^{2} + \left(8 a^{5} + 2 a^{4} + 10 a^{3} + a^{2} + a + 3\right)\cdot 13^{3} + \left(7 a^{5} + 12 a^{4} + 4 a^{3} + 3 a^{2} + 8\right)\cdot 13^{4} + \left(2 a^{5} + 11 a^{4} + 6 a^{3} + 10 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 2 a^{5} + 9 a^{4} + 6 a^{3} + 5 a^{2} + 9 a + 6 + \left(11 a^{5} + 2 a^{3} + 12 a^{2} + 4 a\right)\cdot 13 + \left(10 a^{5} + 9 a^{4} + 12 a^{3} + 11 a^{2} + 6 a + 8\right)\cdot 13^{2} + \left(9 a^{5} + 6 a^{4} + 11 a^{3} + 10 a + 12\right)\cdot 13^{3} + \left(3 a^{4} + 6 a^{3} + 9 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(6 a^{5} + 3 a^{4} + 6 a^{3} + 12 a^{2} + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ a^{5} + 6 a^{3} + 3 a^{2} + 2 a + 11 + \left(8 a^{5} + 10 a^{4} + 10 a^{2} + 4 a + 8\right)\cdot 13 + \left(12 a^{5} + 6 a^{4} + a^{3} + 7 a^{2} + 2 a + 3\right)\cdot 13^{2} + \left(7 a^{5} + 3 a^{4} + 3 a^{3} + 10 a^{2} + a + 10\right)\cdot 13^{3} + \left(4 a^{5} + 10 a^{4} + a^{3} + 10 a + 6\right)\cdot 13^{4} + \left(4 a^{5} + 10 a^{4} + 2 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 4 a^{5} + 12 a^{4} + 3 a^{3} + 2 a^{2} + 6 a + 1 + \left(11 a^{4} + 2 a^{3} + 8 a + 6\right)\cdot 13 + \left(12 a^{5} + a^{4} + 12 a^{3} + 8 a^{2} + 5 a + 7\right)\cdot 13^{2} + \left(a^{4} + 8 a^{3} + 8 a^{2} + 3 a + 9\right)\cdot 13^{3} + \left(12 a^{5} + 8 a^{4} + 10 a^{3} + 11 a^{2} + a + 6\right)\cdot 13^{4} + \left(9 a^{5} + 3 a^{4} + 3 a^{2} + 7 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 6 a^{5} + a^{4} + 5 a^{3} + 2 a^{2} + a + 5 + \left(3 a^{5} + 10 a^{4} + 8 a^{3} + a^{2} + 9 a + 2\right)\cdot 13 + \left(11 a^{5} + 10 a^{4} + 8 a^{3} + 7 a^{2} + 7 a\right)\cdot 13^{2} + \left(12 a^{5} + 9 a^{4} + 4 a^{3} + 6 a^{2} + 2 a + 1\right)\cdot 13^{3} + \left(3 a^{5} + 12 a^{4} + 3 a^{3} + 8 a^{2} + 7 a + 6\right)\cdot 13^{4} + \left(5 a^{5} + 10 a^{4} + 7 a^{3} + 9 a^{2} + 7 a + 9\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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,10)(2,5)(3,6)(4,7)(8,11)(9,12)$ | $-2$ |
| $3$ | $2$ | $(1,5)(2,10)(3,8)(4,9)(6,11)(7,12)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,12)(4,11)(5,9)(6,10)(7,8)$ | $0$ |
| $1$ | $3$ | $(1,11,12)(2,3,4)(5,6,7)(8,9,10)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,12,11)(2,4,3)(5,7,6)(8,10,9)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,12,11)(2,3,4)(5,6,7)(8,10,9)$ | $-1$ |
| $2$ | $3$ | $(2,4,3)(5,7,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,3,4)(5,6,7)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,8,12,10,11,9)(2,6,4,5,3,7)$ | $2 \zeta_{3} + 2$ |
| $1$ | $6$ | $(1,9,11,10,12,8)(2,7,3,5,4,6)$ | $-2 \zeta_{3}$ |
| $2$ | $6$ | $(1,8,12,10,11,9)(2,7,3,5,4,6)$ | $1$ |
| $2$ | $6$ | $(1,10)(2,7,3,5,4,6)(8,11)(9,12)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,10)(2,6,4,5,3,7)(8,11)(9,12)$ | $\zeta_{3}$ |
| $3$ | $6$ | $(1,7,11,5,12,6)(2,9,3,10,4,8)$ | $0$ |
| $3$ | $6$ | $(1,6,12,5,11,7)(2,8,4,10,3,9)$ | $0$ |
| $3$ | $6$ | $(1,2,11,3,12,4)(5,8,6,9,7,10)$ | $0$ |
| $3$ | $6$ | $(1,4,12,3,11,2)(5,10,7,9,6,8)$ | $0$ |