Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(4293\)\(\medspace = 3^{4} \cdot 53 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.347733.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.53.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.227529.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 6x^{4} - 7x^{3} + 3x^{2} - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 8\cdot 37 + 21\cdot 37^{2} + 8\cdot 37^{3} + 3\cdot 37^{4} + 20\cdot 37^{5} + 7\cdot 37^{6} + 37^{7} + 12\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 28\cdot 37 + 15\cdot 37^{2} + 28\cdot 37^{3} + 33\cdot 37^{4} + 16\cdot 37^{5} + 29\cdot 37^{6} + 35\cdot 37^{7} + 24\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 9 + \left(12 a + 15\right)\cdot 37 + \left(23 a + 33\right)\cdot 37^{2} + \left(21 a + 23\right)\cdot 37^{3} + \left(27 a + 29\right)\cdot 37^{4} + \left(25 a + 17\right)\cdot 37^{5} + \left(34 a + 17\right)\cdot 37^{6} + \left(8 a + 36\right)\cdot 37^{7} + \left(19 a + 2\right)\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 27 + \left(a + 12\right)\cdot 37 + \left(33 a + 27\right)\cdot 37^{2} + \left(13 a + 25\right)\cdot 37^{3} + \left(32 a + 34\right)\cdot 37^{4} + \left(2 a + 28\right)\cdot 37^{5} + \left(36 a + 21\right)\cdot 37^{6} + \left(18 a + 35\right)\cdot 37^{7} + \left(27 a + 9\right)\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 11 + \left(35 a + 24\right)\cdot 37 + \left(3 a + 9\right)\cdot 37^{2} + \left(23 a + 11\right)\cdot 37^{3} + \left(4 a + 2\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} + 15\cdot 37^{6} + \left(18 a + 1\right)\cdot 37^{7} + \left(9 a + 27\right)\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 a + 29 + \left(24 a + 21\right)\cdot 37 + \left(13 a + 3\right)\cdot 37^{2} + \left(15 a + 13\right)\cdot 37^{3} + \left(9 a + 7\right)\cdot 37^{4} + \left(11 a + 19\right)\cdot 37^{5} + \left(2 a + 19\right)\cdot 37^{6} + 28 a\cdot 37^{7} + \left(17 a + 34\right)\cdot 37^{8} +O(37^{9})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,2,4,3)$ | $0$ |
| $4$ | $6$ | $(1,3,4,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.