Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(1658944\)\(\medspace = 2^{6} \cdot 7^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.1658944.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.2.1658944.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 5x^{4} - 8x^{2} + 4x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 65 + \left(15 a + 64\right)\cdot 83 + \left(58 a + 53\right)\cdot 83^{2} + \left(23 a + 19\right)\cdot 83^{3} + \left(18 a + 62\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 33 + \left(55 a + 25\right)\cdot 83 + \left(61 a + 17\right)\cdot 83^{2} + \left(66 a + 26\right)\cdot 83^{3} + \left(69 a + 56\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 79 a + 37 + \left(27 a + 76\right)\cdot 83 + \left(21 a + 23\right)\cdot 83^{2} + \left(16 a + 31\right)\cdot 83^{3} + \left(13 a + 59\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 61 a + 4 + \left(67 a + 58\right)\cdot 83 + \left(24 a + 13\right)\cdot 83^{2} + \left(59 a + 68\right)\cdot 83^{3} + \left(64 a + 56\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 21\cdot 83 + 3\cdot 83^{2} + 50\cdot 83^{3} + 79\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 + 3\cdot 83 + 54\cdot 83^{2} + 53\cdot 83^{3} + 17\cdot 83^{4} +O(83^{5})\)
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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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.