Properties

Label 10.14731e4.30t176.1
Dimension 10
Group $S_6$
Conductor $ 14731^{4}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$47090024679574321= 14731^{4} $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{3} - x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 21 + 42\cdot 47 + 46\cdot 47^{2} + 10\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 + 18\cdot 47 + 28\cdot 47^{2} + 9\cdot 47^{3} + 6\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 10 + \left(29 a + 17\right)\cdot 47 + \left(46 a + 29\right)\cdot 47^{2} + \left(12 a + 40\right)\cdot 47^{3} + \left(38 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 a + 11 + \left(7 a + 14\right)\cdot 47 + \left(8 a + 14\right)\cdot 47^{2} + \left(45 a + 12\right)\cdot 47^{3} + \left(26 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 32 a + 40 + \left(17 a + 13\right)\cdot 47 + 46\cdot 47^{2} + \left(34 a + 19\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 a + 1 + \left(39 a + 35\right)\cdot 47 + \left(38 a + 22\right)\cdot 47^{2} + a\cdot 47^{3} + \left(20 a + 39\right)\cdot 47^{4} +O\left(47^{ 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
$c1$
$1$ $1$ $()$ $10$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-2$
$15$ $2$ $(1,2)$ $2$
$45$ $2$ $(1,2)(3,4)$ $-2$
$40$ $3$ $(1,2,3)(4,5,6)$ $1$
$40$ $3$ $(1,2,3)$ $1$
$90$ $4$ $(1,2,3,4)(5,6)$ $0$
$90$ $4$ $(1,2,3,4)$ $0$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $1$
$120$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.