Properties

Label 1.5.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $5$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(5\)
Artin number field: Galois closure of \(\Q(\zeta_{5})\)
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{5}(2,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{4} - x^{3} + x^{2} - x + 1 $.

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 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)$

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.