Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(1011\)\(\medspace = 3 \cdot 337 \) |
Artin field: | Galois closure of 4.0.344454777.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{1011}(485,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 211x^{2} + 1032x + 17628 \) . |
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 7\cdot 13 + 5\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 8\cdot 13^{5} +O(13^{6})\) |
$r_{ 2 }$ | $=$ | \( 4 + 5\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} + 10\cdot 13^{5} +O(13^{6})\) |
$r_{ 3 }$ | $=$ | \( 11 + 9\cdot 13 + 2\cdot 13^{2} + 3\cdot 13^{3} + 2\cdot 13^{4} + 3\cdot 13^{5} +O(13^{6})\) |
$r_{ 4 }$ | $=$ | \( 12 + 3\cdot 13 + 6\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} +O(13^{6})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
$1$ | $4$ | $(1,2,3,4)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,4,3,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.