Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.13436928.4 |
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 6.2.13436928.4 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 6x^{4} - 4x^{3} + 6x^{2} - 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 89 a + 50 + \left(63 a + 149\right)\cdot 167 + \left(34 a + 77\right)\cdot 167^{2} + \left(a + 66\right)\cdot 167^{3} + \left(3 a + 6\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 2 }$ | $=$ | \( 107 a + 68 + \left(105 a + 149\right)\cdot 167 + \left(19 a + 125\right)\cdot 167^{2} + \left(68 a + 63\right)\cdot 167^{3} + \left(78 a + 129\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 3 }$ | $=$ | \( 60 a + 8 + \left(61 a + 148\right)\cdot 167 + \left(147 a + 39\right)\cdot 167^{2} + \left(98 a + 112\right)\cdot 167^{3} + \left(88 a + 139\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 4 }$ | $=$ | \( 101 + 73\cdot 167 + 118\cdot 167^{2} + 26\cdot 167^{3} + 43\cdot 167^{4} +O(167^{5})\) |
$r_{ 5 }$ | $=$ | \( 135 + 23\cdot 167 + 90\cdot 167^{2} + 31\cdot 167^{3} + 7\cdot 167^{4} +O(167^{5})\) |
$r_{ 6 }$ | $=$ | \( 78 a + 139 + \left(103 a + 123\right)\cdot 167 + \left(132 a + 48\right)\cdot 167^{2} + \left(165 a + 33\right)\cdot 167^{3} + \left(163 a + 8\right)\cdot 167^{4} +O(167^{5})\) |
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.