Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(592\)\(\medspace = 2^{4} \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.2368.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.148.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{37})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} - 4x^{2} + 4x + 10 \) . |
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 48 + 21\cdot 149 + 76\cdot 149^{2} + 10\cdot 149^{3} + 149^{4} +O(149^{5})\) |
$r_{ 2 }$ | $=$ | \( 58 + 139\cdot 149 + 38\cdot 149^{2} + 46\cdot 149^{3} + 12\cdot 149^{4} +O(149^{5})\) |
$r_{ 3 }$ | $=$ | \( 65 + 75\cdot 149 + 85\cdot 149^{2} + 24\cdot 149^{3} + 89\cdot 149^{4} +O(149^{5})\) |
$r_{ 4 }$ | $=$ | \( 129 + 61\cdot 149 + 97\cdot 149^{2} + 67\cdot 149^{3} + 46\cdot 149^{4} +O(149^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)$ | $0$ |
$2$ | $2$ | $(1,2)$ | $0$ |
$2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.