Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(318977\)\(\medspace = 37^{2} \cdot 233 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.11802149.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.233.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.11802149.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - 4x^{3} + 6x^{2} + 5x - 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$:
\( x^{2} + 103x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 67 + \left(86 a + 89\right)\cdot 107 + \left(28 a + 22\right)\cdot 107^{2} + \left(64 a + 81\right)\cdot 107^{3} + \left(28 a + 80\right)\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 63 a + 29 + \left(20 a + 70\right)\cdot 107 + \left(78 a + 51\right)\cdot 107^{2} + \left(42 a + 95\right)\cdot 107^{3} + \left(78 a + 23\right)\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 63 a + 73 + \left(67 a + 58\right)\cdot 107 + \left(70 a + 57\right)\cdot 107^{2} + \left(25 a + 37\right)\cdot 107^{3} + \left(100 a + 64\right)\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 4 + \left(39 a + 52\right)\cdot 107 + \left(36 a + 58\right)\cdot 107^{2} + \left(81 a + 69\right)\cdot 107^{3} + \left(6 a + 11\right)\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 78 + 78\cdot 107 + 67\cdot 107^{2} + 42\cdot 107^{3} + 56\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 71 + 78\cdot 107 + 62\cdot 107^{2} + 101\cdot 107^{3} + 83\cdot 107^{4} +O(107^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,6)$ | $2$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,6)$ | $1$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $12$ | $6$ | $(2,6)(3,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.