Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(1001\)\(\medspace = 7 \cdot 11 \cdot 13 \) |
Artin field: | Galois closure of 4.0.13026013.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{1001}(307,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 249x^{2} + 251x + 4943 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 25 + 35\cdot 43 + 11\cdot 43^{2} + 29\cdot 43^{3} + 11\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 30 + 5\cdot 43 + 29\cdot 43^{2} + 37\cdot 43^{3} + 12\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 34 + 4\cdot 43 + 40\cdot 43^{2} + 23\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 41 + 39\cdot 43 + 4\cdot 43^{2} + 38\cdot 43^{3} + 34\cdot 43^{4} +O(43^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.