Properties

Label 2.144.4t3.b
Dimension 2
Group $D_{4}$
Conductor $ 2^{4} \cdot 3^{2}$
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$144= 2^{4} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{4} - 3 x^{2} + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{12})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 7 + 12\cdot 37 + 23\cdot 37^{2} + 25\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 30\cdot 37 + 31\cdot 37^{2} + 18\cdot 37^{3} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 6\cdot 37 + 5\cdot 37^{2} + 18\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 + 24\cdot 37 + 13\cdot 37^{2} + 11\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 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.