Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(2496\)\(\medspace = 2^{6} \cdot 3 \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.4.32448.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.156.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{13})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 10x^{2} + 12 \) . |
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 18 + 47\cdot 61 + 52\cdot 61^{2} + 33\cdot 61^{3} + 58\cdot 61^{4} +O(61^{5})\) |
$r_{ 2 }$ | $=$ | \( 28 + 37\cdot 61 + 41\cdot 61^{2} + 42\cdot 61^{3} + 37\cdot 61^{4} +O(61^{5})\) |
$r_{ 3 }$ | $=$ | \( 33 + 23\cdot 61 + 19\cdot 61^{2} + 18\cdot 61^{3} + 23\cdot 61^{4} +O(61^{5})\) |
$r_{ 4 }$ | $=$ | \( 43 + 13\cdot 61 + 8\cdot 61^{2} + 27\cdot 61^{3} + 2\cdot 61^{4} +O(61^{5})\) |
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.