Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(65\)\(\medspace = 5 \cdot 13 \) |
Artin field: | Galois closure of 4.0.21125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{65}(12,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 16x^{2} - 16x + 61 \) . |
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 12 + 43\cdot 59 + 3\cdot 59^{2} + 23\cdot 59^{3} + 9\cdot 59^{4} +O(59^{5})\) |
$r_{ 2 }$ | $=$ | \( 14 + 10\cdot 59 + 22\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\) |
$r_{ 3 }$ | $=$ | \( 43 + 10\cdot 59 + 27\cdot 59^{2} + 59^{3} + 17\cdot 59^{4} +O(59^{5})\) |
$r_{ 4 }$ | $=$ | \( 50 + 53\cdot 59 + 5\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} +O(59^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.