Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1444\)\(\medspace = 2^{2} \cdot 19^{2} \) |
| Artin stem field: | Galois closure of 6.0.39617584.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.19.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.76.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - 11x^{3} + 18x^{2} + 33x + 83 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 3 + \left(2 a + 8\right)\cdot 31 + \left(10 a + 20\right)\cdot 31^{2} + \left(29 a + 13\right)\cdot 31^{3} + 3 a\cdot 31^{4} + \left(12 a + 14\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 6 + \left(2 a + 18\right)\cdot 31 + \left(14 a + 2\right)\cdot 31^{2} + \left(5 a + 21\right)\cdot 31^{3} + \left(26 a + 2\right)\cdot 31^{4} + \left(a + 12\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 28 + \left(28 a + 16\right)\cdot 31 + \left(20 a + 6\right)\cdot 31^{2} + a\cdot 31^{3} + \left(27 a + 10\right)\cdot 31^{4} + \left(18 a + 3\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + \left(12 a + 22\right)\cdot 31 + \left(16 a + 22\right)\cdot 31^{2} + \left(5 a + 22\right)\cdot 31^{3} + \left(4 a + 13\right)\cdot 31^{4} + \left(18 a + 5\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 14 + \left(28 a + 18\right)\cdot 31 + \left(16 a + 28\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(4 a + 18\right)\cdot 31^{4} + \left(29 a + 20\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 12 + \left(18 a + 9\right)\cdot 31 + \left(14 a + 12\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(26 a + 16\right)\cdot 31^{4} + \left(12 a + 6\right)\cdot 31^{5} +O(31^{6})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |
| $3$ | $6$ | $(1,4,6,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.