Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(195364\)\(\medspace = 2^{2} \cdot 13^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.13284752.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 10x^{2} + 212x + 628 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 80\cdot 89 + 42\cdot 89^{2} + 17\cdot 89^{3} + 28\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 + 32\cdot 89 + 29\cdot 89^{2} + 17\cdot 89^{3} + 67\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 57 + 26\cdot 89 + 89^{2} + 38\cdot 89^{3} + 53\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 61 + 38\cdot 89 + 15\cdot 89^{2} + 16\cdot 89^{3} + 29\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.