Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Artin field: | Galois closure of \(\Q(\zeta_{20})^+\) |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{20}(3,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 5x^{2} + 5 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 6\cdot 19 + 18\cdot 19^{2} + 5\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 + 13\cdot 19 + 6\cdot 19^{2} + 14\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 + 5\cdot 19 + 12\cdot 19^{2} + 4\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 13 + 12\cdot 19 + 13\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
$1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.