Properties

Label 10.2e6_3461e4.30t176.1
Dimension 10
Group $S_6$
Conductor $ 2^{6} \cdot 3461^{4}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$9183037782236224= 2^{6} \cdot 3461^{4} $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - x + 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 $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: $ x^{2} + 145 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 64 + 134\cdot 149 + 21\cdot 149^{2} + 46\cdot 149^{3} + 69\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 65 + 122\cdot 149 + 70\cdot 149^{2} + 106\cdot 149^{3} + 23\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 95 a + 31 + \left(51 a + 53\right)\cdot 149 + \left(27 a + 47\right)\cdot 149^{2} + \left(102 a + 24\right)\cdot 149^{3} + \left(115 a + 42\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 + 47\cdot 149 + 45\cdot 149^{2} + 43\cdot 149^{3} + 107\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 54 a + 113 + \left(97 a + 15\right)\cdot 149 + \left(121 a + 105\right)\cdot 149^{2} + \left(46 a + 107\right)\cdot 149^{3} + \left(33 a + 104\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 + 74\cdot 149 + 7\cdot 149^{2} + 119\cdot 149^{3} + 99\cdot 149^{4} +O\left(149^{ 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.