Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $468= 2^{2} \cdot 3^{2} \cdot 13 $ |
| Artin number field: | Splitting field of $f= x^{12} + x^{10} - 4 x^{9} + x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + 5 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.2e2_13.6t1.1c2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 7 a^{5} + 7 a^{4} + 18 a^{3} + 11 a^{2} + 2 a + 14 + \left(13 a^{5} + 5 a^{4} + a^{3} + 16 a^{2} + 4 a + 1\right)\cdot 19 + \left(7 a^{5} + 7 a^{4} + 18 a^{3} + a^{2} + 5 a + 12\right)\cdot 19^{2} + \left(14 a^{5} + 12 a^{4} + 10 a^{2} + 3 a + 14\right)\cdot 19^{3} + \left(14 a^{5} + 11 a^{4} + 7 a^{3} + 7 a^{2} + 10 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 13 a^{5} + 9 a^{4} + 15 a^{3} + 7 a + \left(11 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 5\right)\cdot 19 + \left(13 a^{5} + 5 a^{4} + 9 a^{3} + a + 7\right)\cdot 19^{2} + \left(8 a^{5} + a^{4} + 12 a^{3} + 17 a^{2} + 12 a + 12\right)\cdot 19^{3} + \left(16 a^{5} + 17 a^{4} + 12 a^{3} + 18 a^{2} + a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 8 a^{5} + 16 a^{4} + 4 a^{3} + 11 a^{2} + 4 a + 2 + \left(2 a^{5} + 13 a^{4} + 15 a^{3} + a^{2} + 10\right)\cdot 19 + \left(7 a^{5} + 9 a^{4} + 8 a^{3} + 15 a^{2} + 5 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 10 a^{4} + 7 a^{3} + 15 a^{2} + 2 a + 14\right)\cdot 19^{3} + \left(a^{5} + a^{4} + 6 a^{3} + 14 a^{2} + 6 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 2 a^{5} + 8 a^{4} + 11 a^{3} + 5 a^{2} + 2 a + 1 + \left(15 a^{5} + 10 a^{4} + 5 a^{3} + 17 a^{2} + 15 a + 13\right)\cdot 19 + \left(9 a^{5} + 2 a^{4} + a^{3} + 18 a^{2} + 14 a + 9\right)\cdot 19^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 15\right)\cdot 19^{3} + \left(5 a^{5} + 3 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 11 a^{5} + 11 a^{4} + 12 a^{3} + a^{2} + 14 a + 3 + \left(8 a^{5} + 9 a^{4} + 17 a^{3} + 15 a^{2} + a + 13\right)\cdot 19 + \left(18 a^{5} + 9 a^{4} + 15 a^{3} + 15 a^{2} + 17 a + 18\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 6 a^{3} + 5 a^{2} + 15 a + 10\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 6 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 15 a^{5} + 6 a^{4} + 10 a^{3} + 11 a + \left(10 a^{5} + 11 a^{4} + 6 a^{3} + 4 a^{2} + 12 a + 16\right)\cdot 19 + \left(15 a^{4} + 6 a^{3} + 16 a^{2} + 4 a + 11\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 9 a^{3} + 13 a^{2} + 7 a + 2\right)\cdot 19^{3} + \left(12 a^{5} + 15 a^{4} + 14 a^{3} + 9 a^{2} + 16 a\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 10 a^{5} + 13 a^{4} + 13 a^{3} + 18 a^{2} + 4 a + 7 + \left(7 a^{5} + 4 a^{4} + 6 a^{3} + 8 a^{2} + 2 a + 11\right)\cdot 19 + \left(12 a^{5} + 5 a^{4} + 12 a^{3} + 4 a^{2} + 10 a + 1\right)\cdot 19^{2} + \left(5 a^{5} + 15 a^{4} + 17 a^{3} + 8 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(17 a^{5} + 3 a^{4} + 15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 13 a^{5} + 15 a^{4} + 13 a^{3} + 16 a^{2} + 17 a + 13 + \left(15 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + a + 3\right)\cdot 19 + \left(10 a^{5} + 15 a^{4} + 5 a^{2} + 2 a + 13\right)\cdot 19^{2} + \left(15 a^{5} + 12 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 2\right)\cdot 19^{3} + \left(3 a^{5} + 9 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 14 a^{5} + 18 a^{4} + a^{3} + 16 a^{2} + 14 a + 7 + \left(2 a^{5} + 2 a^{4} + 14 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 19 + \left(5 a^{5} + 2 a^{4} + 17 a^{3} + 8 a^{2} + 2 a + 4\right)\cdot 19^{2} + \left(9 a^{5} + 17 a^{4} + 17 a^{3} + 16 a^{2} + 3\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 3 a^{3} + 3 a^{2} + 3 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 12 a^{5} + 5 a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 3 + \left(12 a^{5} + 10 a^{4} + 8 a^{3} + 12 a^{2} + 14 a + 3\right)\cdot 19 + \left(2 a^{5} + a^{4} + 15 a^{3} + 6 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(12 a^{5} + 15 a^{4} + 5 a^{3} + 16 a^{2} + 8\right)\cdot 19^{3} + \left(18 a^{5} + 11 a^{4} + 18 a^{3} + 2 a^{2} + 11 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 13 a^{5} + 15 a^{4} + 15 a^{3} + 12 a^{2} + 9 a + 11 + \left(12 a^{5} + 18 a^{4} + 13 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 19 + \left(3 a^{5} + 7 a^{4} + 11 a^{3} + 6 a^{2} + 17 a + 2\right)\cdot 19^{2} + \left(12 a^{5} + 9 a^{4} + 13 a^{3} + 16 a^{2} + 7 a + 11\right)\cdot 19^{3} + \left(17 a^{5} + a^{4} + 18 a^{2} + 12 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 15 a^{5} + 10 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 15 + \left(8 a^{3} + a^{2} + 2 a + 11\right)\cdot 19 + \left(3 a^{5} + 12 a^{4} + 15 a^{3} + 14 a^{2} + 10 a + 10\right)\cdot 19^{2} + \left(a^{5} + 8 a^{4} + a^{3} + 7 a^{2} + 4 a + 14\right)\cdot 19^{3} + \left(7 a^{5} + 6 a^{4} + 13 a^{3} + 9 a^{2} + 12 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,10)(4,9)(7,11)(8,12)$ | $-2$ |
| $3$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,9)(4,6)(5,10)(7,12)(8,11)$ | $0$ |
| $1$ | $3$ | $(1,8,4)(2,10,7)(3,11,6)(5,12,9)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,8)(2,7,10)(3,6,11)(5,9,12)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,4,8)(5,9,12)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,8,4)(5,12,9)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,8)(2,10,7)(3,11,6)(5,9,12)$ | $-1$ |
| $1$ | $6$ | $(1,9,8,5,4,12)(2,11,10,6,7,3)$ | $-2 \zeta_{3}$ |
| $1$ | $6$ | $(1,12,4,5,8,9)(2,3,7,6,10,11)$ | $2 \zeta_{3} + 2$ |
| $2$ | $6$ | $(1,12,4,5,8,9)(2,6)(3,10)(7,11)$ | $\zeta_{3}$ |
| $2$ | $6$ | $(1,9,8,5,4,12)(2,6)(3,10)(7,11)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,12,4,5,8,9)(2,11,10,6,7,3)$ | $1$ |
| $3$ | $6$ | $(1,6,8,3,4,11)(2,12,10,9,7,5)$ | $0$ |
| $3$ | $6$ | $(1,11,4,3,8,6)(2,5,7,9,10,12)$ | $0$ |
| $3$ | $6$ | $(1,10,4,2,8,7)(3,9,6,12,11,5)$ | $0$ |
| $3$ | $6$ | $(1,7,8,2,4,10)(3,5,11,12,6,9)$ | $0$ |