Properties

Label 2.2e5_7.4t3.2
Dimension 2
Group $D_{4}$
Conductor $ 2^{5} \cdot 7 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$224= 2^{5} \cdot 7 $
Artin number field: Splitting field of $f= x^{4} - x^{2} + 2 $ 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 $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 18 + 22\cdot 71 + 50\cdot 71^{2} + 60\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 + 35\cdot 71 + 48\cdot 71^{2} + 11\cdot 71^{3} + 9\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 48 + 35\cdot 71 + 22\cdot 71^{2} + 59\cdot 71^{3} + 61\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 53 + 48\cdot 71 + 20\cdot 71^{2} + 10\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$

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

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

Character values on conjugacy classes

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