Properties

Label 1.2e4.4t1.2c2
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: Odd
Corresponding Dirichlet character: \(\chi_{16}(3,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 4\cdot 7 + 3\cdot 7^{2} + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 + 3\cdot 7 + 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 + 3\cdot 7 + 6\cdot 7^{2} + 5\cdot 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 + 2\cdot 7 + 3\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} +O\left(7^{ 5 }\right)$

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.