Basic invariants
Dimension: | $16$ |
Group: | $S_6$ |
Conductor: | \(609\!\cdots\!121\)\(\medspace = 22291^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.22291.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1252 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.22291.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + x^{4} - x^{3} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \( x^{2} + 101x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 60 a + 91 + \left(85 a + 84\right)\cdot 113 + \left(109 a + 82\right)\cdot 113^{2} + \left(109 a + 88\right)\cdot 113^{3} + \left(39 a + 65\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 2 }$ | $=$ | \( 78 + 98\cdot 113 + 28\cdot 113^{2} + 66\cdot 113^{3} + 94\cdot 113^{4} +O(113^{5})\) |
$r_{ 3 }$ | $=$ | \( 103 + 62\cdot 113 + 82\cdot 113^{2} + 56\cdot 113^{3} + 94\cdot 113^{4} +O(113^{5})\) |
$r_{ 4 }$ | $=$ | \( 53 a + 20 + \left(27 a + 34\right)\cdot 113 + \left(3 a + 71\right)\cdot 113^{2} + \left(3 a + 55\right)\cdot 113^{3} + \left(73 a + 96\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 5 }$ | $=$ | \( 9 a + 26 + \left(47 a + 90\right)\cdot 113 + \left(31 a + 97\right)\cdot 113^{2} + \left(21 a + 36\right)\cdot 113^{3} + \left(69 a + 41\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 6 }$ | $=$ | \( 104 a + 21 + \left(65 a + 81\right)\cdot 113 + \left(81 a + 88\right)\cdot 113^{2} + \left(91 a + 34\right)\cdot 113^{3} + \left(43 a + 59\right)\cdot 113^{4} +O(113^{5})\) |
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 |
$1$ | $1$ | $()$ | $16$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$15$ | $2$ | $(1,2)$ | $0$ |
$45$ | $2$ | $(1,2)(3,4)$ | $0$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
$40$ | $3$ | $(1,2,3)$ | $-2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.