Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(845\)\(\medspace = 5 \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.142805.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.4225.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{3} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 16 + \left(17 a + 24\right)\cdot 31 + \left(17 a + 3\right)\cdot 31^{2} + \left(13 a + 11\right)\cdot 31^{3} + \left(a + 22\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 27 + \left(15 a + 25\right)\cdot 31 + \left(11 a + 14\right)\cdot 31^{2} + 25\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 7\cdot 31 + 19\cdot 31^{2} + 28\cdot 31^{3} + 29\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 17\cdot 31 + 11\cdot 31^{2} + 23\cdot 31^{3} + 26\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 14 + \left(15 a + 16\right)\cdot 31 + \left(19 a + 22\right)\cdot 31^{2} + \left(30 a + 14\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 10 + \left(13 a + 1\right)\cdot 31 + \left(13 a + 21\right)\cdot 31^{2} + \left(17 a + 20\right)\cdot 31^{3} + \left(29 a + 11\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
| $4$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.