Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Artin field: | Galois closure of 4.0.72000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{120}(107,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 30x^{2} + 180 \) . |
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 35\cdot 59 + 25\cdot 59^{2} + 13\cdot 59^{3} + 7\cdot 59^{4} +O(59^{5})\)
$r_{ 2 }$ |
$=$ |
\( 16 + 32\cdot 59 + 35\cdot 59^{2} + 48\cdot 59^{3} + 31\cdot 59^{4} +O(59^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 43 + 26\cdot 59 + 23\cdot 59^{2} + 10\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 56 + 23\cdot 59 + 33\cdot 59^{2} + 45\cdot 59^{3} + 51\cdot 59^{4} +O(59^{5})\)
| |
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.