Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(17525\)\(\medspace = 5^{2} \cdot 701 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.87625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.701.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.87625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + x^{3} - 3x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 9\cdot 29 + 10\cdot 29^{2} + 13\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 18 + \left(27 a + 11\right)\cdot 29 + \left(22 a + 9\right)\cdot 29^{2} + \left(14 a + 11\right)\cdot 29^{3} + \left(17 a + 10\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 8 + \left(a + 8\right)\cdot 29 + \left(6 a + 9\right)\cdot 29^{2} + \left(14 a + 4\right)\cdot 29^{3} + \left(11 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 28\cdot 29 + 13\cdot 29^{2} + 17\cdot 29^{3} + 28\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + \left(12 a + 22\right)\cdot 29 + \left(7 a + 9\right)\cdot 29^{2} + \left(12 a + 22\right)\cdot 29^{3} + \left(16 a + 8\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 13 + \left(16 a + 7\right)\cdot 29 + \left(21 a + 5\right)\cdot 29^{2} + \left(16 a + 18\right)\cdot 29^{3} + \left(12 a + 20\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $2$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.