Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(19097\)\(\medspace = 13^{2} \cdot 113 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.248261.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.113.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.248261.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 4x^{4} - 2x^{3} - 2x^{2} + 4x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 19\cdot 61 + 2\cdot 61^{2} + 2\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 47 + \left(44 a + 16\right)\cdot 61 + \left(a + 56\right)\cdot 61^{2} + \left(8 a + 15\right)\cdot 61^{3} + \left(35 a + 57\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 a + 11 + \left(30 a + 24\right)\cdot 61 + \left(48 a + 56\right)\cdot 61^{2} + \left(57 a + 50\right)\cdot 61^{3} + \left(14 a + 5\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 54 + \left(30 a + 11\right)\cdot 61 + \left(12 a + 13\right)\cdot 61^{2} + \left(3 a + 60\right)\cdot 61^{3} + \left(46 a + 23\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 + 41\cdot 61^{2} + 31\cdot 61^{3} + 39\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 a + 59 + \left(16 a + 48\right)\cdot 61 + \left(59 a + 13\right)\cdot 61^{2} + \left(52 a + 22\right)\cdot 61^{3} + \left(25 a + 23\right)\cdot 61^{4} +O(61^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.