Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(79895\)\(\medspace = 5 \cdot 19 \cdot 29^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.399475.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.95.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} + 16x^{2} - 15x - 995 \) . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 8\cdot 11 + 8\cdot 11^{2} + 11^{3} + 11^{4} + 3\cdot 11^{5} + 11^{6} + 8\cdot 11^{7} +O(11^{8})\) |
$r_{ 2 }$ | $=$ | \( 4 + 8\cdot 11 + 3\cdot 11^{2} + 3\cdot 11^{3} + 2\cdot 11^{4} + 9\cdot 11^{5} + 5\cdot 11^{6} + 9\cdot 11^{7} +O(11^{8})\) |
$r_{ 3 }$ | $=$ | \( 8 + 2\cdot 11 + 7\cdot 11^{2} + 7\cdot 11^{3} + 8\cdot 11^{4} + 11^{5} + 5\cdot 11^{6} + 11^{7} +O(11^{8})\) |
$r_{ 4 }$ | $=$ | \( 9 + 2\cdot 11 + 2\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} + 7\cdot 11^{5} + 9\cdot 11^{6} + 2\cdot 11^{7} +O(11^{8})\) |
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.