Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(10345347\)\(\medspace = 3^{3} \cdot 619^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.50139.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.50139.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - x^{3} - x^{2} + 2x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 27 + 3\cdot 31 + 9\cdot 31^{2} + 17\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 17\cdot 31 + 10\cdot 31^{2} + 8\cdot 31^{3} + 12\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 25 + \left(27 a + 4\right)\cdot 31 + \left(14 a + 13\right)\cdot 31^{2} + \left(28 a + 12\right)\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 13 + 31 + \left(25 a + 13\right)\cdot 31^{2} + \left(21 a + 6\right)\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 5 + \left(3 a + 8\right)\cdot 31 + \left(16 a + 15\right)\cdot 31^{2} + \left(2 a + 23\right)\cdot 31^{3} + \left(30 a + 11\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 27 + \left(30 a + 25\right)\cdot 31 + 5 a\cdot 31^{2} + \left(9 a + 25\right)\cdot 31^{3} + \left(30 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,5)$ | $0$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
| $4$ | $3$ | $(1,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.