Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(145\)\(\medspace = 5 \cdot 29 \) |
Artin field: | Galois closure of 4.0.609725.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{145}(99,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 33x^{2} - 107x + 139 \) . |
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 5\cdot 13 + 13^{2} + 7\cdot 13^{3} + 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 + 10\cdot 13 + 9\cdot 13^{3} + 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( 3 + 4\cdot 13^{2} + 4\cdot 13^{3} + 7\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 + 10\cdot 13 + 6\cdot 13^{2} + 5\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\) |
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.