Properties

Label 2.2e3_3_11.4t3.2
Dimension 2
Group $D_{4}$
Conductor $ 2^{3} \cdot 3 \cdot 11 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$264= 2^{3} \cdot 3 \cdot 11 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 5 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 15\cdot 17 + 5\cdot 17^{2} + 2\cdot 17^{3} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 16\cdot 17^{2} + 13\cdot 17^{3} + 7\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 + 14\cdot 17 + 9\cdot 17^{2} + 6\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 14 + 3\cdot 17 + 2\cdot 17^{2} + 11\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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