Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(2525\)\(\medspace = 5^{2} \cdot 101 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.12625.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.101.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{101})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 11x^{2} - 6x + 36 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 15\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 3 + 6\cdot 19 + 6\cdot 19^{2} + 4\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 + 15\cdot 19 + 11\cdot 19^{2} + 7\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 11 + 19 + 16\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} +O(19^{5})\) |
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.