Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(4753\)\(\medspace = 7^{2} \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.232897.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.97.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.461041.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 7x^{4} - 9x^{3} + 7x^{2} - 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 43 + 25\cdot 43^{2} + 26\cdot 43^{4} + 37\cdot 43^{5} + 23\cdot 43^{6} + 22\cdot 43^{7} + 5\cdot 43^{8} +O(43^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 a + 2 + \left(27 a + 6\right)\cdot 43 + \left(6 a + 32\right)\cdot 43^{2} + \left(21 a + 35\right)\cdot 43^{3} + \left(18 a + 22\right)\cdot 43^{4} + \left(19 a + 42\right)\cdot 43^{5} + \left(19 a + 42\right)\cdot 43^{6} + \left(41 a + 31\right)\cdot 43^{7} + \left(14 a + 34\right)\cdot 43^{8} +O(43^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 12 + \left(30 a + 16\right)\cdot 43 + \left(12 a + 30\right)\cdot 43^{2} + \left(4 a + 25\right)\cdot 43^{3} + \left(9 a + 40\right)\cdot 43^{4} + \left(34 a + 8\right)\cdot 43^{5} + \left(38 a + 19\right)\cdot 43^{6} + \left(22 a + 29\right)\cdot 43^{7} + \left(24 a + 20\right)\cdot 43^{8} +O(43^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 42 + \left(15 a + 36\right)\cdot 43 + \left(36 a + 10\right)\cdot 43^{2} + \left(21 a + 7\right)\cdot 43^{3} + \left(24 a + 20\right)\cdot 43^{4} + 23 a\cdot 43^{5} + 23 a\cdot 43^{6} + \left(a + 11\right)\cdot 43^{7} + \left(28 a + 8\right)\cdot 43^{8} +O(43^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 41\cdot 43 + 17\cdot 43^{2} + 42\cdot 43^{3} + 16\cdot 43^{4} + 5\cdot 43^{5} + 19\cdot 43^{6} + 20\cdot 43^{7} + 37\cdot 43^{8} +O(43^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 32 + \left(12 a + 26\right)\cdot 43 + \left(30 a + 12\right)\cdot 43^{2} + \left(38 a + 17\right)\cdot 43^{3} + \left(33 a + 2\right)\cdot 43^{4} + \left(8 a + 34\right)\cdot 43^{5} + \left(4 a + 23\right)\cdot 43^{6} + \left(20 a + 13\right)\cdot 43^{7} + \left(18 a + 22\right)\cdot 43^{8} +O(43^{9})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,4)$ | $-1$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $6$ | $(1,6,4,5,3,2)$ | $0$ |
| $4$ | $6$ | $(1,2,3,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.