Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
Artin field: | Galois closure of 4.0.256000.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{80}(13,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 20x^{2} + 10 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 19\cdot 31 + 8\cdot 31^{2} + 2\cdot 31^{3} + 18\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 14 + 15\cdot 31 + 30\cdot 31^{2} + 5\cdot 31^{3} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 17 + 15\cdot 31 + 25\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 30 + 11\cdot 31 + 22\cdot 31^{2} + 28\cdot 31^{3} + 12\cdot 31^{4} +O(31^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
$1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.