Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(1520\)\(\medspace = 2^{4} \cdot 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.8.333621760000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| 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 $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 12\cdot 61 + 52\cdot 61^{2} + 52\cdot 61^{3} + 31\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 57\cdot 61 + 35\cdot 61^{2} + 57\cdot 61^{3} + 42\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 49\cdot 61 + 31\cdot 61^{2} + 19\cdot 61^{3} + 32\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 21\cdot 61 + 9\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 47\cdot 61 + 2\cdot 61^{2} + 51\cdot 61^{3} + 32\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 45 + 36\cdot 61 + 19\cdot 61^{2} + 44\cdot 61^{3} + 31\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 53 + 2\cdot 61 + 25\cdot 61^{2} + 18\cdot 61^{3} + 34\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 55 + 16\cdot 61 + 6\cdot 61^{2} + 37\cdot 61^{3} + 11\cdot 61^{4} +O(61^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,6,5,7)(2,3,8,4)$ | $0$ |