Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(86051769821\)\(\medspace = 17^{4} \cdot 101^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.29189.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | even |
Determinant: | 1.101.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.29189.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + x^{4} + x^{3} - 2x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: \( x^{2} + 108x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 97 a + 59 + \left(40 a + 17\right)\cdot 109 + \left(86 a + 40\right)\cdot 109^{2} + \left(2 a + 24\right)\cdot 109^{3} + \left(15 a + 40\right)\cdot 109^{4} +O(109^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 a + 80 + \left(a + 76\right)\cdot 109 + \left(92 a + 36\right)\cdot 109^{2} + \left(3 a + 18\right)\cdot 109^{3} + \left(82 a + 75\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 12 a + 47 + \left(68 a + 70\right)\cdot 109 + \left(22 a + 85\right)\cdot 109^{2} + \left(106 a + 49\right)\cdot 109^{3} + \left(93 a + 52\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 93 a + 37 + \left(33 a + 74\right)\cdot 109 + \left(3 a + 33\right)\cdot 109^{2} + \left(82 a + 58\right)\cdot 109^{3} + \left(51 a + 72\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 104 a + 85 + \left(107 a + 72\right)\cdot 109 + \left(16 a + 18\right)\cdot 109^{2} + \left(105 a + 39\right)\cdot 109^{3} + \left(26 a + 44\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 16 a + 21 + \left(75 a + 15\right)\cdot 109 + \left(105 a + 3\right)\cdot 109^{2} + \left(26 a + 28\right)\cdot 109^{3} + \left(57 a + 42\right)\cdot 109^{4} +O(109^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $-1$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.