Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
Artin field: | Galois closure of 4.0.316368.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{156}(83,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 39x^{2} + 117 \) . |
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 16\cdot 17^{2} + 10\cdot 17^{3} + 3\cdot 17^{4} + 16\cdot 17^{5} +O(17^{6})\)
$r_{ 2 }$ |
$=$ |
\( 5 + 3\cdot 17 + 17^{2} + 11\cdot 17^{5} +O(17^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 12 + 13\cdot 17 + 15\cdot 17^{2} + 16\cdot 17^{3} + 16\cdot 17^{4} + 5\cdot 17^{5} +O(17^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 15 + 16\cdot 17 + 6\cdot 17^{3} + 13\cdot 17^{4} +O(17^{6})\)
| |
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.