Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(845471929\)\(\medspace = 29077^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.29077.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T183 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.29077.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} - x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a + 129 + \left(55 a + 128\right)\cdot 131 + \left(78 a + 32\right)\cdot 131^{2} + \left(19 a + 108\right)\cdot 131^{3} + \left(57 a + 50\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 2 }$ | $=$ | \( 123 a + 30 + \left(75 a + 79\right)\cdot 131 + \left(52 a + 29\right)\cdot 131^{2} + \left(111 a + 108\right)\cdot 131^{3} + \left(73 a + 128\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 3 }$ | $=$ | \( 111 a + 49 + \left(54 a + 105\right)\cdot 131 + \left(61 a + 111\right)\cdot 131^{2} + \left(18 a + 53\right)\cdot 131^{3} + \left(13 a + 4\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 4 }$ | $=$ | \( 55 a + 64 + \left(70 a + 81\right)\cdot 131 + \left(109 a + 101\right)\cdot 131^{2} + \left(126 a + 25\right)\cdot 131^{3} + \left(24 a + 33\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 5 }$ | $=$ | \( 76 a + 22 + \left(60 a + 46\right)\cdot 131 + \left(21 a + 76\right)\cdot 131^{2} + \left(4 a + 30\right)\cdot 131^{3} + \left(106 a + 6\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 6 }$ | $=$ | \( 20 a + 100 + \left(76 a + 82\right)\cdot 131 + \left(69 a + 40\right)\cdot 131^{2} + \left(112 a + 66\right)\cdot 131^{3} + \left(117 a + 38\right)\cdot 131^{4} +O(131^{5})\) |
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.