Properties

Label 1.2e4.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 2^{4}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$16= 2^{4} $
Artin number field: Splitting field of $f= x^{4} - 4 x^{2} + 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{16}(13,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 5 + 3\cdot 17 + 14\cdot 17^{2} + 12\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 2 } &= 8 + 17 + 9\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 3 } &= 9 + 15\cdot 17 + 16\cdot 17^{2} + 7\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 4 } &= 12 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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