Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(55263\)\(\medspace = 3 \cdot 13^{2} \cdot 109 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.165789.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.327.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{109})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 61x^{2} + 55x + 1153 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 17\cdot 31 + 9\cdot 31^{2} + 22\cdot 31^{3} + 24\cdot 31^{4} + 30\cdot 31^{5} +O(31^{6})\) |
$r_{ 2 }$ | $=$ | \( 4 + 14\cdot 31 + 20\cdot 31^{2} + 16\cdot 31^{3} + 24\cdot 31^{4} + 19\cdot 31^{5} +O(31^{6})\) |
$r_{ 3 }$ | $=$ | \( 5 + 28\cdot 31 + 14\cdot 31^{2} + 17\cdot 31^{3} + 24\cdot 31^{4} + 29\cdot 31^{5} +O(31^{6})\) |
$r_{ 4 }$ | $=$ | \( 22 + 2\cdot 31 + 17\cdot 31^{2} + 5\cdot 31^{3} + 19\cdot 31^{4} + 12\cdot 31^{5} +O(31^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,3)$ | $0$ |
$2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.