Properties

Label 2.2e2_29.6t3.1
Dimension 2
Group $D_{6}$
Conductor $ 2^{2} \cdot 29 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$116= 2^{2} \cdot 29 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{3} + x^{2} + 2 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 15 a + 6 + \left(12 a + 8\right)\cdot 17 + \left(16 a + 10\right)\cdot 17^{2} + \left(6 a + 13\right)\cdot 17^{3} + \left(16 a + 9\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 4 + \left(4 a + 6\right)\cdot 17 + 14\cdot 17^{2} + \left(10 a + 3\right)\cdot 17^{3} + 2\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 10 + \left(3 a + 5\right)\cdot 17 + 12 a\cdot 17^{2} + \left(6 a + 15\right)\cdot 17^{3} + \left(a + 3\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 + 2\cdot 17 + 9\cdot 17^{2} + 16\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 9 a + 1 + \left(13 a + 1\right)\cdot 17 + \left(4 a + 9\right)\cdot 17^{2} + \left(10 a + 9\right)\cdot 17^{3} + \left(15 a + 15\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 + 10\cdot 17 + 7\cdot 17^{2} + 9\cdot 17^{3} + 14\cdot 17^{4} +O\left(17^{ 5 }\right)$

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

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

Character values on conjugacy classes

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