Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(912\)\(\medspace = 2^{4} \cdot 3 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.2736.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.228.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{19})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 8x^{2} + 19 \) . |
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 76\cdot 79 + 72\cdot 79^{2} + 39\cdot 79^{3} + 60\cdot 79^{4} +O(79^{5})\) |
$r_{ 2 }$ | $=$ | \( 29 + 59\cdot 79 + 67\cdot 79^{2} + 18\cdot 79^{3} + 33\cdot 79^{4} +O(79^{5})\) |
$r_{ 3 }$ | $=$ | \( 50 + 19\cdot 79 + 11\cdot 79^{2} + 60\cdot 79^{3} + 45\cdot 79^{4} +O(79^{5})\) |
$r_{ 4 }$ | $=$ | \( 73 + 2\cdot 79 + 6\cdot 79^{2} + 39\cdot 79^{3} + 18\cdot 79^{4} +O(79^{5})\) |
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.