Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(309291125\)\(\medspace = 5^{3} \cdot 11^{4} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.1546455625.1 |
| 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 6.2.1546455625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 11x^{4} + 23x^{3} + 66x^{2} - 300x - 829 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 19 a + 16 + \left(20 a + 18\right)\cdot 29 + \left(21 a + 1\right)\cdot 29^{2} + \left(a + 28\right)\cdot 29^{3} + \left(7 a + 14\right)\cdot 29^{4} + \left(18 a + 8\right)\cdot 29^{5} + \left(17 a + 20\right)\cdot 29^{6} + \left(4 a + 9\right)\cdot 29^{7} + \left(18 a + 19\right)\cdot 29^{8} + \left(9 a + 17\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 2\cdot 29 + 20\cdot 29^{2} + 15\cdot 29^{3} + 8\cdot 29^{4} + 13\cdot 29^{5} + 6\cdot 29^{6} + 4\cdot 29^{7} + 13\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 8 + \left(24 a + 26\right)\cdot 29 + \left(11 a + 6\right)\cdot 29^{2} + 28 a\cdot 29^{3} + \left(3 a + 12\right)\cdot 29^{4} + \left(27 a + 9\right)\cdot 29^{5} + \left(4 a + 24\right)\cdot 29^{6} + \left(4 a + 14\right)\cdot 29^{7} + \left(20 a + 22\right)\cdot 29^{8} + \left(22 a + 18\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 24 + \left(8 a + 15\right)\cdot 29 + \left(7 a + 2\right)\cdot 29^{2} + \left(27 a + 15\right)\cdot 29^{3} + \left(21 a + 19\right)\cdot 29^{4} + \left(10 a + 5\right)\cdot 29^{5} + \left(11 a + 3\right)\cdot 29^{6} + \left(24 a + 15\right)\cdot 29^{7} + \left(10 a + 18\right)\cdot 29^{8} + \left(19 a + 18\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 + 2\cdot 29 + 14\cdot 29^{2} + 13\cdot 29^{3} + 28\cdot 29^{4} + 24\cdot 29^{5} + 10\cdot 29^{6} + 12\cdot 29^{7} + 10\cdot 29^{8} + 23\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 5 + \left(4 a + 21\right)\cdot 29 + \left(17 a + 12\right)\cdot 29^{2} + 14\cdot 29^{3} + \left(25 a + 3\right)\cdot 29^{4} + \left(a + 25\right)\cdot 29^{5} + \left(24 a + 21\right)\cdot 29^{6} + \left(24 a + 1\right)\cdot 29^{7} + \left(8 a + 3\right)\cdot 29^{8} + \left(6 a + 25\right)\cdot 29^{9} +O(29^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $9$ | $4$ | $(1,3,2,5)(4,6)$ | $0$ |
| $9$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.