Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Artin field: | Galois closure of 4.0.100352.5 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{112}(69,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 28x^{2} + 98 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 8 + 21\cdot 41 + 31\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$ |
$=$ |
\( 20 + 20\cdot 41 + 12\cdot 41^{2} + 5\cdot 41^{3} + 11\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 21 + 20\cdot 41 + 28\cdot 41^{2} + 35\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 33 + 19\cdot 41 + 9\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O(41^{5})\)
| |
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.