Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $1296= 2^{4} \cdot 3^{4} $ |
| Artin number field: | Splitting field of $f= x^{12} - 2 x^{9} + 2 x^{6} - 4 x^{3} + 4 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.3e2.6t1.1c2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 11 a^{5} + 6 a^{3} + 8 a^{2} + 2 a + 11 + \left(15 a^{5} + 10 a^{4} + 2 a^{3} + 18 a^{2} + 14 a + 14\right)\cdot 19 + \left(a^{5} + a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 11\right)\cdot 19^{2} + \left(9 a^{5} + 9 a^{4} + 11 a^{3} + 17 a^{2} + 12 a + 2\right)\cdot 19^{3} + \left(8 a^{5} + 6 a^{4} + 11 a^{3} + 12 a^{2} + 16 a + 2\right)\cdot 19^{4} + \left(2 a^{5} + 5 a^{4} + 18 a^{3} + 14 a^{2} + 10 a + 6\right)\cdot 19^{5} + \left(13 a^{5} + 8 a^{4} + 17 a^{3} + 3 a^{2} + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 15 a^{5} + 7 a^{4} + 18 a^{3} + 15 a^{2} + 10 a + 16 + \left(7 a^{4} + a^{3} + 4 a^{2} + 8 a + 5\right)\cdot 19 + \left(7 a^{5} + 2 a^{4} + a^{3} + 9 a^{2} + 15 a + 8\right)\cdot 19^{2} + \left(9 a^{5} + 3 a^{4} + 9 a^{3} + 6 a^{2} + 10\right)\cdot 19^{3} + \left(a^{5} + 7 a^{4} + 14 a^{3} + 8 a^{2} + 4 a + 18\right)\cdot 19^{4} + \left(2 a^{5} + 14 a^{4} + 4 a^{3} + 15 a^{2} + 11 a + 7\right)\cdot 19^{5} + \left(14 a^{5} + 11 a^{4} + 12 a^{3} + 4 a^{2} + 5 a + 13\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 3 a^{5} + 9 a^{4} + 15 a^{3} + 3 a^{2} + 2 a + 7 + \left(2 a^{5} + 7 a^{4} + 2 a^{3} + 18 a^{2} + 8 a + 8\right)\cdot 19 + \left(2 a^{5} + 13 a^{4} + 16 a^{3} + 11 a^{2} + 8 a + 11\right)\cdot 19^{2} + \left(18 a^{5} + 2 a^{4} + 9 a^{2} + 15 a + 17\right)\cdot 19^{3} + \left(13 a^{5} + 17 a^{4} + 17 a^{3} + 18 a^{2} + 2 a + 9\right)\cdot 19^{4} + \left(8 a^{5} + 18 a^{4} + a^{3} + 9 a^{2} + 7 a + 17\right)\cdot 19^{5} + \left(7 a^{5} + 10 a^{4} + a^{3} + 6 a^{2} + 11 a + 12\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 5 a^{5} + a^{3} + 14 a^{2} + 13 a + 5 + \left(5 a^{5} + 8 a^{4} + 11 a^{3} + 6 a^{2} + a + 8\right)\cdot 19 + \left(10 a^{5} + 6 a^{4} + 6 a^{3} + 12 a^{2} + 14 a + 15\right)\cdot 19^{2} + \left(12 a^{5} + 11 a^{4} + 4 a^{3} + 18 a^{2} + 15 a + 12\right)\cdot 19^{3} + \left(9 a^{5} + 10 a^{4} + 11 a^{3} + 3 a^{2} + 13 a + 13\right)\cdot 19^{4} + \left(15 a^{5} + 5 a^{4} + 4 a^{3} + 3 a^{2} + 12 a + 11\right)\cdot 19^{5} + \left(17 a^{5} + 4 a^{4} + 18 a^{3} + 2 a + 4\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ a^{5} + 4 a^{3} + 18 a^{2} + 14 a + 1 + \left(8 a^{5} + 13 a^{4} + 11 a^{3} + a^{2} + 14 a + 1\right)\cdot 19 + \left(8 a^{4} + 3 a^{3} + 17 a^{2} + 17\right)\cdot 19^{2} + \left(10 a^{5} + 4 a^{4} + 8 a^{3} + 13 a^{2} + 3 a + 11\right)\cdot 19^{3} + \left(10 a^{5} + 2 a^{4} + 10 a^{3} + 9 a^{2} + 11 a + 14\right)\cdot 19^{4} + \left(3 a^{5} + 18 a^{4} + 17 a^{3} + 12 a^{2} + 10 a + 1\right)\cdot 19^{5} + \left(14 a^{5} + 3 a^{4} + 17 a^{3} + 15 a^{2} + 3 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 10 a^{5} + 11 a^{4} + 12 a^{3} + 10 a^{2} + 13 a + 17 + \left(2 a^{5} + 4 a^{4} + 17 a^{3} + 11 a^{2} + 14\right)\cdot 19 + \left(14 a^{5} + 5 a^{4} + 10 a^{3} + 13 a^{2} + 2 a + 10\right)\cdot 19^{2} + \left(5 a^{5} + 14 a^{4} + 17 a^{3} + 8 a^{2} + 12 a + 10\right)\cdot 19^{3} + \left(6 a^{5} + 6 a^{4} + 15 a^{3} + 7 a^{2} + 6 a + 18\right)\cdot 19^{4} + \left(12 a^{5} + 2 a^{4} + 12 a^{3} + 2 a^{2} + 10 a + 16\right)\cdot 19^{5} + \left(9 a^{5} + 8 a^{4} + 9 a^{3} + a^{2} + 5 a\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 2 a^{5} + 6 a^{4} + 10 a^{3} + 2 a^{2} + 14 a + 11 + \left(3 a^{5} + 16 a^{4} + 16 a^{3} + a^{2} + 10 a + 11\right)\cdot 19 + \left(11 a^{5} + 2 a^{4} + 12 a^{3} + 16 a^{2} + 4 a + 7\right)\cdot 19^{2} + \left(2 a^{5} + 6 a^{4} + a^{3} + 12 a^{2} + 6 a + 11\right)\cdot 19^{3} + \left(7 a^{5} + 16 a^{4} + 13 a^{3} + 18 a^{2} + 8 a + 2\right)\cdot 19^{4} + \left(16 a^{5} + 7 a^{4} + 15 a^{3} + 11 a^{2} + 16 a + 16\right)\cdot 19^{5} + \left(17 a^{5} + a^{4} + 12 a^{3} + 17 a^{2} + 18 a + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 16 a^{5} + 7 a^{3} + 3 a^{2} + 15 a + 16 + \left(16 a^{5} + 18 a^{4} + 16 a^{3} + 10 a^{2} + 16 a + 18\right)\cdot 19 + \left(14 a^{5} + 12 a^{4} + 9 a^{3} + 5 a\right)\cdot 19^{2} + \left(12 a^{5} + 3 a^{4} + 16 a^{3} + 2 a^{2} + 10 a + 18\right)\cdot 19^{3} + \left(9 a^{5} + 4 a^{4} + 9 a^{2} + 7 a + 1\right)\cdot 19^{4} + \left(18 a^{5} + 13 a^{3} + 15 a^{2} + 15 a\right)\cdot 19^{5} + \left(16 a^{5} + 15 a^{4} + 5 a^{3} + a^{2} + 8 a + 12\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 7 a^{5} + 9 a^{3} + 12 a^{2} + 3 a + 7 + \left(14 a^{5} + 15 a^{4} + 5 a^{3} + 17 a^{2} + 9 a + 3\right)\cdot 19 + \left(16 a^{5} + 8 a^{4} + 12 a^{3} + 8 a^{2} + a + 9\right)\cdot 19^{2} + \left(18 a^{5} + 5 a^{4} + 18 a^{3} + 6 a^{2} + 3 a + 4\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 15 a^{3} + 15 a^{2} + 10 a + 2\right)\cdot 19^{4} + \left(12 a^{5} + 14 a^{4} + a^{3} + 10 a^{2} + 16 a + 11\right)\cdot 19^{5} + \left(10 a^{5} + 6 a^{4} + 2 a^{3} + 18 a^{2} + 14 a + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 13 a^{5} + a^{4} + 8 a^{3} + 13 a^{2} + 15 a + 5 + \left(15 a^{5} + 7 a^{4} + 18 a^{3} + 2 a^{2} + 9 a + 17\right)\cdot 19 + \left(16 a^{5} + 11 a^{4} + 6 a^{3} + 15 a^{2} + a + 18\right)\cdot 19^{2} + \left(3 a^{5} + a^{4} + 11 a^{3} + 3 a^{2} + 6 a + 16\right)\cdot 19^{3} + \left(11 a^{5} + 5 a^{4} + 7 a^{3} + 3 a^{2} + 8 a\right)\cdot 19^{4} + \left(4 a^{5} + 2 a^{4} + a^{3} + a^{2} + 16 a + 13\right)\cdot 19^{5} + \left(14 a^{5} + 18 a^{4} + 16 a^{3} + 13 a^{2} + 7 a + 4\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 14 a^{5} + 4 a^{4} + 13 a^{3} + 14 a^{2} + 3 a + 1 + \left(13 a^{5} + 14 a^{4} + 18 a^{3} + 18 a^{2} + 18\right)\cdot 19 + \left(5 a^{5} + 2 a^{4} + 8 a^{3} + 9 a^{2} + 6 a + 18\right)\cdot 19^{2} + \left(17 a^{5} + 10 a^{4} + 16 a^{3} + 15 a^{2} + 16 a + 8\right)\cdot 19^{3} + \left(16 a^{5} + 4 a^{4} + 7 a^{3} + 7 a + 6\right)\cdot 19^{4} + \left(12 a^{5} + 11 a^{4} + a^{3} + 16 a^{2} + 14 a + 4\right)\cdot 19^{5} + \left(12 a^{5} + 6 a^{4} + 5 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 17 a^{5} + 11 a^{3} + 2 a^{2} + 10 a + 17 + \left(15 a^{5} + 12 a^{4} + 10 a^{3} + 2 a^{2} + 10\right)\cdot 19 + \left(12 a^{5} + 18 a^{4} + 2 a^{3} + 6 a^{2} + 18 a + 2\right)\cdot 19^{2} + \left(12 a^{5} + 3 a^{4} + 17 a^{3} + 17 a^{2} + 11 a + 7\right)\cdot 19^{3} + \left(18 a^{5} + 4 a^{4} + 6 a^{3} + 5 a^{2} + 16 a + 3\right)\cdot 19^{4} + \left(3 a^{5} + 13 a^{4} + a^{3} + 9 a + 7\right)\cdot 19^{5} + \left(3 a^{5} + 18 a^{4} + 14 a^{3} + 17 a^{2} + 7 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-2$ |
| $3$ | $2$ | $(1,8)(2,7)(3,6)(4,5)(9,12)(10,11)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,12)(4,11)(5,10)(6,9)(7,8)$ | $0$ |
| $1$ | $3$ | $(1,9,5)(2,6,10)(3,11,7)(4,8,12)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,9)(2,10,6)(3,7,11)(4,12,8)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,9,5)(2,10,6)(3,11,7)(4,12,8)$ | $-1$ |
| $2$ | $3$ | $(2,6,10)(4,8,12)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,10,6)(4,12,8)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,11,9,7,5,3)(2,4,6,8,10,12)$ | $-2 \zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,7,9,11)(2,12,10,8,6,4)$ | $2 \zeta_{3} + 2$ |
| $2$ | $6$ | $(1,3,5,7,9,11)(2,4,6,8,10,12)$ | $1$ |
| $2$ | $6$ | $(1,7)(2,12,10,8,6,4)(3,9)(5,11)$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,7)(2,4,6,8,10,12)(3,9)(5,11)$ | $-\zeta_{3} - 1$ |
| $3$ | $6$ | $(1,8,9,12,5,4)(2,3,6,11,10,7)$ | $0$ |
| $3$ | $6$ | $(1,4,5,12,9,8)(2,7,10,11,6,3)$ | $0$ |
| $3$ | $6$ | $(1,2,9,6,5,10)(3,12,11,4,7,8)$ | $0$ |
| $3$ | $6$ | $(1,10,5,6,9,2)(3,8,7,4,11,12)$ | $0$ |