Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(2556125\)\(\medspace = 5^{3} \cdot 11^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.12780625.1 |
| 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 6.2.12780625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 6x^{4} + 19x^{3} - x^{2} - 37x + 31 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 4 + 6\cdot 29 + 26\cdot 29^{2} + 22\cdot 29^{3} + 21\cdot 29^{4} + 9\cdot 29^{5} + 6\cdot 29^{6} + 9\cdot 29^{7} + 7\cdot 29^{8} + 7\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 14 + \left(26 a + 8\right)\cdot 29 + \left(10 a + 20\right)\cdot 29^{2} + \left(23 a + 28\right)\cdot 29^{3} + \left(11 a + 3\right)\cdot 29^{4} + \left(28 a + 2\right)\cdot 29^{5} + \left(5 a + 16\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(3 a + 11\right)\cdot 29^{8} + \left(16 a + 9\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 27 + \left(2 a + 5\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(5 a + 18\right)\cdot 29^{3} + \left(17 a + 10\right)\cdot 29^{4} + 16\cdot 29^{5} + \left(23 a + 17\right)\cdot 29^{6} + \left(28 a + 2\right)\cdot 29^{7} + \left(25 a + 26\right)\cdot 29^{8} + \left(12 a + 28\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 14\cdot 29 + 18\cdot 29^{2} + 10\cdot 29^{3} + 14\cdot 29^{4} + 10\cdot 29^{5} + 24\cdot 29^{6} + 18\cdot 29^{7} + 20\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 7 + \left(12 a + 16\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(5 a + 9\right)\cdot 29^{3} + \left(a + 3\right)\cdot 29^{4} + \left(16 a + 28\right)\cdot 29^{5} + \left(19 a + 13\right)\cdot 29^{6} + \left(17 a + 4\right)\cdot 29^{7} + \left(18 a + 2\right)\cdot 29^{8} + \left(13 a + 15\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 19 + \left(16 a + 6\right)\cdot 29 + \left(18 a + 21\right)\cdot 29^{2} + \left(23 a + 25\right)\cdot 29^{3} + \left(27 a + 3\right)\cdot 29^{4} + \left(12 a + 20\right)\cdot 29^{5} + \left(9 a + 8\right)\cdot 29^{6} + \left(11 a + 15\right)\cdot 29^{7} + \left(10 a + 19\right)\cdot 29^{8} + \left(15 a + 6\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)$ | $1$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $9$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $9$ | $4$ | $(1,3,5,2)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.