Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.9800.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 11\cdot 113 + 96\cdot 113^{2} + 105\cdot 113^{3} + 87\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 + 38\cdot 113 + 96\cdot 113^{2} + 16\cdot 113^{3} + 44\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 75 + 52\cdot 113 + 102\cdot 113^{2} + 43\cdot 113^{3} + 99\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 109 + 10\cdot 113 + 44\cdot 113^{2} + 59\cdot 113^{3} + 107\cdot 113^{4} +O(113^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |