Properties

Label 1.5.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 5 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$5 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{5}(3,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 2 + 10\cdot 11 + 4\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 2 } &= 6 + 8\cdot 11 + 5\cdot 11^{2} + 9\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 3 } &= 7 + 3\cdot 11 + 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 4 } &= 8 + 10\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} +O\left(11^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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