Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(2476745\)\(\medspace = 5 \cdot 19 \cdot 29^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.1458802805.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.2945.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{589})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 414x^{2} - 1928x + 10607 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 6\cdot 79 + 42\cdot 79^{2} + 5\cdot 79^{3} + 36\cdot 79^{4} + 76\cdot 79^{5} + 17\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 59\cdot 79 + 71\cdot 79^{2} + 75\cdot 79^{3} + 43\cdot 79^{4} + 73\cdot 79^{5} + 62\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 62\cdot 79 + 31\cdot 79^{2} + 70\cdot 79^{3} + 4\cdot 79^{4} + 76\cdot 79^{5} + 61\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 + 30\cdot 79 + 12\cdot 79^{2} + 6\cdot 79^{3} + 73\cdot 79^{4} + 10\cdot 79^{5} + 15\cdot 79^{6} +O(79^{7})\)
|
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(2,3)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.