Properties

Label 2.5_11.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 11 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$55= 5 \cdot 11 $
Artin number field: Splitting field of $f=x^{4} - x^{3} + x^{2} + x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.5_11.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 262175886 +O\left(59^{ 5 }\right) \\ r_{ 2 } &= 249399050 +O\left(59^{ 5 }\right) \\ r_{ 3 } &= -171153360 +O\left(59^{ 5 }\right) \\ r_{ 4 } &= -340421575 +O\left(59^{ 5 }\right) \\ \end{aligned}\]

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.