Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(97\) |
Artin field: | Galois closure of 4.4.912673.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{97}(75,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 36x^{2} - 91x - 61 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 14\cdot 43 + 27\cdot 43^{2} + 8\cdot 43^{3} + 8\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 22 + 21\cdot 43 + 41\cdot 43^{2} + 41\cdot 43^{3} + 42\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 23 + 24\cdot 43 + 22\cdot 43^{2} + 23\cdot 43^{3} + 10\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 32 + 25\cdot 43 + 37\cdot 43^{2} + 11\cdot 43^{3} + 24\cdot 43^{4} +O(43^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
$1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.