Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(2667\)\(\medspace = 3 \cdot 7 \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.2370963.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.2667.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{889})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 11x^{2} - 192 \) . |
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 63 + 34\cdot 223 + 103\cdot 223^{2} + 201\cdot 223^{3} + 64\cdot 223^{4} +O(223^{5})\) |
$r_{ 2 }$ | $=$ | \( 82 + 115\cdot 223 + 182\cdot 223^{2} + 75\cdot 223^{3} + 97\cdot 223^{4} +O(223^{5})\) |
$r_{ 3 }$ | $=$ | \( 141 + 107\cdot 223 + 40\cdot 223^{2} + 147\cdot 223^{3} + 125\cdot 223^{4} +O(223^{5})\) |
$r_{ 4 }$ | $=$ | \( 160 + 188\cdot 223 + 119\cdot 223^{2} + 21\cdot 223^{3} + 158\cdot 223^{4} +O(223^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.