Properties

Label 2.2e4_3e2.4t3.3c1
Dimension 2
Group $D_4$
Conductor $ 2^{4} \cdot 3^{2}$
Root number 1
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 $f=x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} - 4 x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= -27517075 +O\left(37^{ 5 }\right) \\ r_{ 2 } &= -4469845 +O\left(37^{ 5 }\right) \\ r_{ 3 } &= 16879863 +O\left(37^{ 5 }\right) \\ r_{ 4 } &= 9240549 +O\left(37^{ 5 }\right) \\ r_{ 5 } &= 3791005 +O\left(37^{ 5 }\right) \\ r_{ 6 } &= 7820129 +O\left(37^{ 5 }\right) \\ r_{ 7 } &= -27204697 +O\left(37^{ 5 }\right) \\ r_{ 8 } &= 21460073 +O\left(37^{ 5 }\right) \\ \end{aligned}\]

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,2)(3,5)(4,7)(6,8)$
$(1,3,6,4)(2,7,8,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,6)(2,8)(3,4)(5,7)$$-2$
$2$$2$$(1,2)(3,5)(4,7)(6,8)$$0$
$2$$2$$(1,7)(2,3)(4,8)(5,6)$$0$
$2$$4$$(1,3,6,4)(2,7,8,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.