Properties

Label 2.2e3_17.4t3.4c1
Dimension 2
Group $D_4$
Conductor $ 2^{3} \cdot 17 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$136= 2^{3} \cdot 17 $
Artin number field: Splitting field of $f=x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 2$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 4T3
Parity: Even
Determinant: 1.2e3_17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 66567240 +O\left(47^{ 5 }\right) \\ r_{ 2 } &= -16614578 +O\left(47^{ 5 }\right) \\ r_{ 3 } &= 16614579 +O\left(47^{ 5 }\right) \\ r_{ 4 } &= -66567239 +O\left(47^{ 5 }\right) \\ \end{aligned}\]

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 value
$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.