Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(475\)\(\medspace = 5^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of \(\Q(\sqrt{-5 +2 \sqrt{-95}})\) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 51 + 114\cdot 131 + 91\cdot 131^{2} + 98\cdot 131^{3} + 106\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 56 + 8\cdot 131 + 30\cdot 131^{2} + 112\cdot 131^{3} + 76\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 76 + 122\cdot 131 + 100\cdot 131^{2} + 18\cdot 131^{3} + 54\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 81 + 16\cdot 131 + 39\cdot 131^{2} + 32\cdot 131^{3} + 24\cdot 131^{4} +O(131^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |