Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.4400.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 29\cdot 31 + 26\cdot 31^{2} + 16\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 10 + 7\cdot 31 + 10\cdot 31^{2} + 19\cdot 31^{3} + 21\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 21 + 23\cdot 31 + 20\cdot 31^{2} + 11\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 26 + 31 + 4\cdot 31^{2} + 14\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(2,3)$ | $0$ |
$2$ | $4$ | $(1,2,4,3)$ | $0$ |