Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(13381632\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.608256.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.12.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.608256.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - 3x^{2} + 2x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 8 + \left(36 a + 37\right)\cdot 59 + \left(24 a + 55\right)\cdot 59^{2} + \left(7 a + 11\right)\cdot 59^{3} + \left(14 a + 10\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 43\cdot 59 + 51\cdot 59^{2} + 49\cdot 59^{3} + 58\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 + 36\cdot 59 + 49\cdot 59^{2} + 33\cdot 59^{3} + 44\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 a + 21 + \left(22 a + 1\right)\cdot 59 + \left(34 a + 44\right)\cdot 59^{2} + \left(51 a + 53\right)\cdot 59^{3} + \left(44 a + 16\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + \left(32 a + 47\right)\cdot 59 + 6 a\cdot 59^{2} + \left(55 a + 19\right)\cdot 59^{3} + 41 a\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 a + 9 + \left(26 a + 11\right)\cdot 59 + \left(52 a + 34\right)\cdot 59^{2} + \left(3 a + 8\right)\cdot 59^{3} + \left(17 a + 46\right)\cdot 59^{4} +O(59^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ |
| $4$ | $3$ | $(1,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $1$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.