Properties

Label 1.2e4.4t1.1
Dimension 1
Group $C_4$
Conductor $ 2^{4}$
Frobenius-Schur indicator 0

Related objects

Learn more about

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

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 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)$

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 values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,4)(2,3)$ $-1$ $-1$
$1$ $4$ $(1,3,4,2)$ $\zeta_{4}$ $-\zeta_{4}$
$1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.