Properties

Label 2.5_19.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$95= 5 \cdot 19 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 3 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.5_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 16 + 124\cdot 131 + 5\cdot 131^{2} + 69\cdot 131^{3} + 23\cdot 131^{4} +O\left(131^{ 5 }\right) \\ r_{ 2 } &= 76 + 90\cdot 131 + 39\cdot 131^{2} + 120\cdot 131^{3} + 60\cdot 131^{4} +O\left(131^{ 5 }\right) \\ r_{ 3 } &= 81 + 81\cdot 131 + 68\cdot 131^{2} + 88\cdot 131^{3} + 106\cdot 131^{4} +O\left(131^{ 5 }\right) \\ r_{ 4 } &= 90 + 96\cdot 131 + 16\cdot 131^{2} + 115\cdot 131^{3} + 70\cdot 131^{4} +O\left(131^{ 5 }\right) \\ \end{aligned}\]

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.