Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Artin field: | Galois closure of 4.0.8000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{40}(37,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 10x^{2} + 20 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)
$r_{ 2 }$ |
$=$ |
\( 9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
$1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.