Properties

Label 5.163_191.6t16.1
Dimension 5
Group $S_6$
Conductor $ 163 \cdot 191 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$31133= 163 \cdot 191 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{4} - x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 10 + 46\cdot 97 + 94\cdot 97^{2} + 64\cdot 97^{3} + 28\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 a + 33 + \left(28 a + 6\right)\cdot 97 + \left(12 a + 23\right)\cdot 97^{2} + \left(83 a + 45\right)\cdot 97^{3} + \left(11 a + 77\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 a + 27 + \left(5 a + 2\right)\cdot 97 + \left(77 a + 2\right)\cdot 97^{2} + \left(81 a + 72\right)\cdot 97^{3} + \left(32 a + 35\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 33 + 31\cdot 97 + 90\cdot 97^{2} + 12\cdot 97^{3} + 59\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 76 a + 48 + \left(91 a + 83\right)\cdot 97 + \left(19 a + 73\right)\cdot 97^{2} + \left(15 a + 76\right)\cdot 97^{3} + \left(64 a + 83\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 87 a + 43 + \left(68 a + 24\right)\cdot 97 + \left(84 a + 7\right)\cdot 97^{2} + \left(13 a + 19\right)\cdot 97^{3} + \left(85 a + 6\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.