Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(28400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 71 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.142000.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-71})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 2\cdot 29 + 3\cdot 29^{2} + 10\cdot 29^{3} + 23\cdot 29^{5} +O(29^{6})\) |
$r_{ 2 }$ | $=$ | \( 5 + 19\cdot 29 + 19\cdot 29^{2} + 23\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} +O(29^{6})\) |
$r_{ 3 }$ | $=$ | \( 24 + 9\cdot 29 + 9\cdot 29^{2} + 5\cdot 29^{3} + 15\cdot 29^{4} + 4\cdot 29^{5} +O(29^{6})\) |
$r_{ 4 }$ | $=$ | \( 26 + 26\cdot 29 + 25\cdot 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} + 5\cdot 29^{5} +O(29^{6})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |