Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(100352\)\(\medspace = 2^{11} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.802816.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:C_4$ |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $C_3^2:C_4$ |
| Projective stem field: | Galois closure of 6.2.802816.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 6x^{2} - 4x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 a + 21 + \left(27 a + 24\right)\cdot 31^{2} + \left(5 a + 29\right)\cdot 31^{3} + \left(28 a + 8\right)\cdot 31^{4} + \left(27 a + 11\right)\cdot 31^{5} + \left(23 a + 11\right)\cdot 31^{6} + \left(7 a + 17\right)\cdot 31^{7} + \left(25 a + 3\right)\cdot 31^{8} + \left(14 a + 2\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 18\cdot 31 + 25\cdot 31^{2} + 30\cdot 31^{3} + 16\cdot 31^{4} + 20\cdot 31^{5} + 27\cdot 31^{6} + 25\cdot 31^{7} + 20\cdot 31^{8} + 29\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 30 + \left(14 a + 29\right)\cdot 31 + \left(20 a + 1\right)\cdot 31^{2} + \left(9 a + 24\right)\cdot 31^{3} + \left(16 a + 6\right)\cdot 31^{4} + \left(7 a + 7\right)\cdot 31^{5} + \left(20 a + 5\right)\cdot 31^{6} + \left(18 a + 11\right)\cdot 31^{7} + \left(3 a + 26\right)\cdot 31^{8} + \left(8 a + 16\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 9 + \left(30 a + 8\right)\cdot 31 + \left(3 a + 15\right)\cdot 31^{2} + \left(25 a + 14\right)\cdot 31^{3} + \left(2 a + 28\right)\cdot 31^{4} + \left(3 a + 7\right)\cdot 31^{5} + 7 a\cdot 31^{6} + \left(23 a + 9\right)\cdot 31^{7} + \left(5 a + 15\right)\cdot 31^{8} + \left(16 a + 6\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 31 + 29\cdot 31^{2} + 31^{3} + 2\cdot 31^{4} + 9\cdot 31^{5} + 10\cdot 31^{6} + 31^{7} + 12\cdot 31^{8} + 8\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 18 + \left(16 a + 3\right)\cdot 31 + \left(10 a + 28\right)\cdot 31^{2} + \left(21 a + 22\right)\cdot 31^{3} + \left(14 a + 29\right)\cdot 31^{4} + \left(23 a + 5\right)\cdot 31^{5} + \left(10 a + 7\right)\cdot 31^{6} + \left(12 a + 28\right)\cdot 31^{7} + \left(27 a + 14\right)\cdot 31^{8} + \left(22 a + 29\right)\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $9$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
| $9$ | $4$ | $(1,3,2,5)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.