Properties

Label 2.3_37.4t3.2
Dimension 2
Group $D_{4}$
Conductor $ 3 \cdot 37 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$111= 3 \cdot 37 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 2 x^{2} + 3 $ 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 $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 45 + 77\cdot 127 + 15\cdot 127^{2} + 79\cdot 127^{3} + 89\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 50 + 20\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 117\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 58 + 67\cdot 127 + 109\cdot 127^{2} + 95\cdot 127^{3} + 38\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 102 + 88\cdot 127 + 101\cdot 127^{2} + 57\cdot 127^{3} + 8\cdot 127^{4} +O\left(127^{ 5 }\right)$

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

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

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.