Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(16\)\(\medspace = 2^{4} \) |
Artin field: | Galois closure of \(\Q(\zeta_{16})^+\) |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{16}(5,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 4x^{2} + 2 \) . |
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 3\cdot 17 + 14\cdot 17^{2} + 12\cdot 17^{3} + 15\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 + 17 + 9\cdot 17^{3} + 15\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 + 15\cdot 17 + 16\cdot 17^{2} + 7\cdot 17^{3} + 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 12 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 17^{4} +O(17^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action 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.