Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(181027383501050641\)\(\medspace = 20627^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.20627.1 |
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 6.0.20627.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} - 2x^{3} + 2x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: \( x^{2} + 274x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O(277^{5})\)
$r_{ 2 }$ |
$=$ |
\( 230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O(277^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O(277^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O(277^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O(277^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O(277^{5})\)
| |
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.