Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(6025\)\(\medspace = 5^{2} \cdot 241 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.69987605.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.241.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.69987605.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 15x^{3} + 15x^{2} - 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 15 a + 24 + \left(22 a + 23\right)\cdot 29 + \left(14 a + 1\right)\cdot 29^{2} + \left(23 a + 4\right)\cdot 29^{3} + 3 a\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 10\cdot 29 + 22\cdot 29^{2} + 19\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 9 + \left(10 a + 16\right)\cdot 29 + \left(20 a + 15\right)\cdot 29^{2} + \left(17 a + 28\right)\cdot 29^{3} + \left(3 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 12 + \left(6 a + 5\right)\cdot 29 + \left(14 a + 24\right)\cdot 29^{2} + \left(5 a + 19\right)\cdot 29^{3} + \left(25 a + 24\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 15 + \left(18 a + 2\right)\cdot 29 + \left(8 a + 20\right)\cdot 29^{2} + \left(11 a + 9\right)\cdot 29^{3} + \left(25 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 + 28\cdot 29 + 2\cdot 29^{2} + 5\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,6,5)$ | $-1$ |
| $12$ | $6$ | $(1,4,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.