Properties

Label 10.2e12_101e6.30t176.1
Dimension 10
Group $S_6$
Conductor $ 2^{12} \cdot 101^{6}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$4347986536861696= 2^{12} \cdot 101^{6} $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - x^{3} + 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 $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 29 a + 41 + \left(10 a + 26\right)\cdot 109 + \left(96 a + 10\right)\cdot 109^{2} + \left(60 a + 6\right)\cdot 109^{3} + \left(81 a + 90\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 a + 69 + \left(25 a + 10\right)\cdot 109 + \left(98 a + 48\right)\cdot 109^{2} + \left(21 a + 83\right)\cdot 109^{3} + \left(20 a + 27\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 96 a + 29 + \left(92 a + 87\right)\cdot 109 + \left(63 a + 94\right)\cdot 109^{2} + \left(32 a + 90\right)\cdot 109^{3} + \left(84 a + 64\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 75 a + 103 + \left(83 a + 1\right)\cdot 109 + \left(10 a + 12\right)\cdot 109^{2} + \left(87 a + 7\right)\cdot 109^{3} + \left(88 a + 26\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 13 a + 16 + \left(16 a + 84\right)\cdot 109 + \left(45 a + 65\right)\cdot 109^{2} + \left(76 a + 59\right)\cdot 109^{3} + \left(24 a + 7\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 80 a + 70 + \left(98 a + 7\right)\cdot 109 + \left(12 a + 96\right)\cdot 109^{2} + \left(48 a + 79\right)\cdot 109^{3} + \left(27 a + 1\right)\cdot 109^{4} +O\left(109^{ 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.