Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(13\) |
Artin field: | Galois closure of 4.0.2197.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{13}(8,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 2x^{2} + 4x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 20\cdot 29 + 7\cdot 29^{2} + 13\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 15 + 25\cdot 29 + 4\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 17 + 28\cdot 29 + 20\cdot 29^{2} + 6\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 24 + 12\cdot 29 + 24\cdot 29^{2} + 18\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
$1$ | $4$ | $(1,2,3,4)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,4,3,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.