Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(5187\)\(\medspace = 3 \cdot 7 \cdot 13 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.8968323.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.5187.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{1729})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{2} - 432 \) . |
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 14 + 61\cdot 67 + 54\cdot 67^{2} + 7\cdot 67^{3} + 27\cdot 67^{4} +O(67^{5})\) |
$r_{ 2 }$ | $=$ | \( 26 + 12\cdot 67 + 66\cdot 67^{2} + 53\cdot 67^{3} + 36\cdot 67^{4} +O(67^{5})\) |
$r_{ 3 }$ | $=$ | \( 41 + 54\cdot 67 + 13\cdot 67^{3} + 30\cdot 67^{4} +O(67^{5})\) |
$r_{ 4 }$ | $=$ | \( 53 + 5\cdot 67 + 12\cdot 67^{2} + 59\cdot 67^{3} + 39\cdot 67^{4} +O(67^{5})\) |
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.