Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(1042568000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 19^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.14440000.2 |
| 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 6.2.14440000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} - 12x^{3} - 24x^{2} - 20x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 6\cdot 11 + 5\cdot 11^{2} + 4\cdot 11^{3} + 11^{4} + 6\cdot 11^{5} + 11^{6} + 3\cdot 11^{7} + 11^{8} + 10\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 5 + \left(7 a + 9\right)\cdot 11 + \left(2 a + 6\right)\cdot 11^{2} + \left(7 a + 2\right)\cdot 11^{3} + \left(5 a + 5\right)\cdot 11^{4} + \left(10 a + 9\right)\cdot 11^{5} + 3\cdot 11^{6} + \left(9 a + 3\right)\cdot 11^{7} + \left(3 a + 5\right)\cdot 11^{8} + \left(6 a + 1\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 10\cdot 11 + 10\cdot 11^{2} + 4\cdot 11^{4} + 6\cdot 11^{6} + 9\cdot 11^{7} + 8\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + \left(5 a + 6\right)\cdot 11 + \left(3 a + 6\right)\cdot 11^{2} + \left(5 a + 5\right)\cdot 11^{3} + \left(6 a + 1\right)\cdot 11^{4} + \left(4 a + 7\right)\cdot 11^{5} + \left(7 a + 10\right)\cdot 11^{6} + \left(2 a + 3\right)\cdot 11^{7} + \left(7 a + 6\right)\cdot 11^{8} + \left(3 a + 7\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 1 + \left(5 a + 2\right)\cdot 11 + \left(7 a + 4\right)\cdot 11^{2} + \left(5 a + 1\right)\cdot 11^{3} + 4 a\cdot 11^{4} + \left(6 a + 8\right)\cdot 11^{5} + \left(3 a + 2\right)\cdot 11^{6} + \left(8 a + 7\right)\cdot 11^{7} + \left(3 a + 10\right)\cdot 11^{8} + \left(7 a + 3\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 8 + \left(3 a + 9\right)\cdot 11 + \left(8 a + 9\right)\cdot 11^{2} + \left(3 a + 6\right)\cdot 11^{3} + \left(5 a + 9\right)\cdot 11^{4} + 11^{5} + \left(10 a + 8\right)\cdot 11^{6} + \left(a + 5\right)\cdot 11^{7} + 7 a\cdot 11^{8} + \left(4 a + 1\right)\cdot 11^{9} +O(11^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(3,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
| $9$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
| $9$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.