Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(36107\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.36107.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.36107.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.36107.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + x^{4} - x^{2} + x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 57 a + 61 + \left(110 a + 39\right)\cdot 131 + \left(68 a + 97\right)\cdot 131^{2} + \left(2 a + 75\right)\cdot 131^{3} + \left(80 a + 56\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 2 }$ | $=$ | \( 74 a + 27 + \left(20 a + 31\right)\cdot 131 + 62 a\cdot 131^{2} + \left(128 a + 17\right)\cdot 131^{3} + \left(50 a + 112\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 3 }$ | $=$ | \( 62 a + 109 + \left(56 a + 108\right)\cdot 131 + \left(97 a + 94\right)\cdot 131^{2} + \left(29 a + 76\right)\cdot 131^{3} + \left(53 a + 99\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 4 }$ | $=$ | \( 69 a + 95 + \left(74 a + 10\right)\cdot 131 + \left(33 a + 35\right)\cdot 131^{2} + \left(101 a + 98\right)\cdot 131^{3} + \left(77 a + 20\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 + 92\cdot 131 + 97\cdot 131^{2} + 96\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\) |
$r_{ 6 }$ | $=$ | \( 79 + 110\cdot 131 + 67\cdot 131^{2} + 28\cdot 131^{3} + 12\cdot 131^{4} +O(131^{5})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.