Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(920\)\(\medspace = 2^{3} \cdot 5 \cdot 23 \) |
Artin field: | Galois closure of 4.0.4232000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{920}(643,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 230x^{2} + 10580 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ |
\( 1 + 12\cdot 19 + 8\cdot 19^{2} + 12\cdot 19^{3} + 10\cdot 19^{4} + 7\cdot 19^{5} + 10\cdot 19^{6} +O(19^{7})\)
$r_{ 2 }$ |
$=$ |
\( 4 + 12\cdot 19 + 11\cdot 19^{2} + 11\cdot 19^{3} + 6\cdot 19^{4} + 14\cdot 19^{5} + 8\cdot 19^{6} +O(19^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 15 + 6\cdot 19 + 7\cdot 19^{2} + 7\cdot 19^{3} + 12\cdot 19^{4} + 4\cdot 19^{5} + 10\cdot 19^{6} +O(19^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 18 + 6\cdot 19 + 10\cdot 19^{2} + 6\cdot 19^{3} + 8\cdot 19^{4} + 11\cdot 19^{5} + 8\cdot 19^{6} +O(19^{7})\)
| |
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.