Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(987840\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.57624000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.57624000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} + 4x^{3} - 9x^{2} - 3x + 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 79 + 52\cdot 79^{2} + 28\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 65\cdot 79 + 23\cdot 79^{2} + 26\cdot 79^{3} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 31 + \left(70 a + 1\right)\cdot 79 + \left(44 a + 1\right)\cdot 79^{2} + \left(10 a + 57\right)\cdot 79^{3} + \left(9 a + 27\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 2 + \left(42 a + 53\right)\cdot 79 + \left(61 a + 3\right)\cdot 79^{2} + \left(40 a + 36\right)\cdot 79^{3} + \left(20 a + 8\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 a + 64 + \left(8 a + 38\right)\cdot 79 + \left(34 a + 54\right)\cdot 79^{2} + \left(68 a + 22\right)\cdot 79^{3} + \left(69 a + 26\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 58 a + 23 + \left(36 a + 74\right)\cdot 79 + \left(17 a + 22\right)\cdot 79^{2} + \left(38 a + 15\right)\cdot 79^{3} + \left(58 a + 67\right)\cdot 79^{4} +O(79^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $4$ | $3$ | $(2,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,6,5)$ | $-1$ |
| $12$ | $6$ | $(1,4,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.