Properties

Label 2.2e2_17.4t3.2c1
Dimension 2
Group $D_4$
Conductor $ 2^{2} \cdot 17 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$68= 2^{2} \cdot 17 $
Artin number field: Splitting field of $f=x^{4} + x^{2} - 2 x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 4T3
Parity: Odd
Determinant: 1.2e2_17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 37617284 +O\left(53^{ 5 }\right) \\ r_{ 2 } &= -67317570 +O\left(53^{ 5 }\right) \\ r_{ 3 } &= -14568939 +O\left(53^{ 5 }\right) \\ r_{ 4 } &= 44269225 +O\left(53^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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