Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(3438880839109836961\)\(\medspace = 43063^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.43063.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.43063.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{3} + x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 61\cdot 89 + 17\cdot 89^{2} + 6\cdot 89^{3} + 40\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 71 a + 24 + \left(26 a + 33\right)\cdot 89 + \left(82 a + 59\right)\cdot 89^{2} + \left(49 a + 16\right)\cdot 89^{3} + \left(11 a + 75\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 44 + 72\cdot 89 + 53\cdot 89^{2} + 30\cdot 89^{3} + 2\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 33\cdot 89 + 27\cdot 89^{2} + 7\cdot 89^{3} + 79\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 49 + 5\cdot 89 + 34\cdot 89^{2} + 11\cdot 89^{3} + 53\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 76 + \left(62 a + 60\right)\cdot 89 + \left(6 a + 74\right)\cdot 89^{2} + \left(39 a + 16\right)\cdot 89^{3} + \left(77 a + 17\right)\cdot 89^{4} +O(89^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.