Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(2312000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.11560000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:C_4$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $C_3^2:C_4$ |
| Projective stem field: | Galois closure of 6.2.11560000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 4x^{4} - 7x^{3} - 16x^{2} - 21x - 11 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 7 + 18 a\cdot 29 + \left(4 a + 2\right)\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(7 a + 18\right)\cdot 29^{4} + \left(21 a + 16\right)\cdot 29^{5} + 11\cdot 29^{6} + \left(8 a + 17\right)\cdot 29^{7} + \left(20 a + 1\right)\cdot 29^{8} + \left(23 a + 20\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 18 + \left(10 a + 25\right)\cdot 29 + \left(24 a + 6\right)\cdot 29^{2} + a\cdot 29^{3} + \left(21 a + 2\right)\cdot 29^{4} + \left(7 a + 28\right)\cdot 29^{5} + \left(28 a + 22\right)\cdot 29^{6} + \left(20 a + 27\right)\cdot 29^{7} + \left(8 a + 7\right)\cdot 29^{8} + \left(5 a + 2\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 24\cdot 29 + 24\cdot 29^{2} + 13\cdot 29^{3} + 23\cdot 29^{4} + 14\cdot 29^{5} + 22\cdot 29^{6} + 12\cdot 29^{7} + 26\cdot 29^{8} + 15\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + \left(19 a + 19\right)\cdot 29 + \left(a + 18\right)\cdot 29^{2} + \left(18 a + 6\right)\cdot 29^{3} + \left(17 a + 20\right)\cdot 29^{4} + \left(9 a + 25\right)\cdot 29^{5} + \left(22 a + 2\right)\cdot 29^{6} + \left(6 a + 17\right)\cdot 29^{7} + \left(5 a + 22\right)\cdot 29^{8} + \left(20 a + 1\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 13 + \left(9 a + 10\right)\cdot 29 + \left(27 a + 7\right)\cdot 29^{2} + \left(10 a + 8\right)\cdot 29^{3} + \left(11 a + 3\right)\cdot 29^{4} + \left(19 a + 27\right)\cdot 29^{5} + \left(6 a + 17\right)\cdot 29^{6} + \left(22 a + 28\right)\cdot 29^{7} + \left(23 a + 12\right)\cdot 29^{8} + \left(8 a + 10\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 + 7\cdot 29 + 27\cdot 29^{2} + 14\cdot 29^{3} + 19\cdot 29^{4} + 3\cdot 29^{5} + 9\cdot 29^{6} + 12\cdot 29^{7} + 15\cdot 29^{8} + 7\cdot 29^{9} +O(29^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $-2$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $9$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $9$ | $4$ | $(1,4,2,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.