Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.11943936.1 |
| 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 6.2.11943936.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - 4x^{3} + 9x^{2} - 12x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 6\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} + 2\cdot 7^{5} + 7^{6} + 7^{7} + 4\cdot 7^{8} + 7^{9} +O(7^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 5 + \left(5 a + 1\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + 2\cdot 7^{3} + \left(a + 2\right)\cdot 7^{4} + \left(3 a + 1\right)\cdot 7^{5} + \left(4 a + 2\right)\cdot 7^{6} + \left(3 a + 3\right)\cdot 7^{7} + \left(4 a + 4\right)\cdot 7^{8} + \left(2 a + 3\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 7 + 7^{2} + 7^{3} + 7^{4} + 5\cdot 7^{5} + 5\cdot 7^{6} + 2\cdot 7^{7} + 3\cdot 7^{8} + 3\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 6 + \left(a + 5\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} + \left(3 a + 3\right)\cdot 7^{5} + \left(2 a + 3\right)\cdot 7^{6} + \left(3 a + 2\right)\cdot 7^{7} + \left(2 a + 5\right)\cdot 7^{8} + \left(4 a + 1\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 2 + \left(6 a + 1\right)\cdot 7 + \left(3 a + 4\right)\cdot 7^{2} + \left(4 a + 2\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} + 6 a\cdot 7^{5} + \left(2 a + 6\right)\cdot 7^{6} + \left(a + 2\right)\cdot 7^{7} + 2\cdot 7^{8} + \left(4 a + 3\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 6 + 3\cdot 7 + \left(3 a + 1\right)\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(a + 3\right)\cdot 7^{4} + 7^{5} + \left(4 a + 2\right)\cdot 7^{6} + \left(5 a + 1\right)\cdot 7^{7} + \left(6 a + 1\right)\cdot 7^{8} + 2 a\cdot 7^{9} +O(7^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $9$ | $4$ | $(1,3,2,5)(4,6)$ | $0$ |
| $9$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.