Properties

Label 10.2e28_3e16.30t176.1
Dimension 10
Group $S_6$
Conductor $ 2^{28} \cdot 3^{16}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$11555266180939776= 2^{28} \cdot 3^{16} $
Artin number field: Splitting field of $f= x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6 $ 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 $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 125 + 13\cdot 167 + 71\cdot 167^{2} + 8\cdot 167^{3} + 52\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 141 + 151\cdot 167 + 21\cdot 167^{2} + 13\cdot 167^{3} + 133\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 67 a + 128 + \left(164 a + 103\right)\cdot 167 + \left(112 a + 44\right)\cdot 167^{2} + \left(59 a + 111\right)\cdot 167^{3} + \left(a + 25\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 a + 28 + \left(2 a + 34\right)\cdot 167 + \left(54 a + 160\right)\cdot 167^{2} + \left(107 a + 57\right)\cdot 167^{3} + \left(165 a + 134\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 131 a + 141 + \left(44 a + 141\right)\cdot 167 + \left(132 a + 57\right)\cdot 167^{2} + \left(58 a + 108\right)\cdot 167^{3} + \left(17 a + 98\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 36 a + 105 + \left(122 a + 55\right)\cdot 167 + \left(34 a + 145\right)\cdot 167^{2} + \left(108 a + 34\right)\cdot 167^{3} + \left(149 a + 57\right)\cdot 167^{4} +O\left(167^{ 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.