Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(43063\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.43063.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.43063.2t1.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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.