Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.8000.1 |
| 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 \(\Q(i, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 4x^{2} + 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 + 54\cdot 101 + 68\cdot 101^{2} + 71\cdot 101^{3} + 95\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 77 + 46\cdot 101 + 82\cdot 101^{2} + 97\cdot 101^{3} + 39\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 80 + 17\cdot 101 + 5\cdot 101^{2} + 87\cdot 101^{3} + 82\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 89 + 82\cdot 101 + 45\cdot 101^{2} + 46\cdot 101^{3} + 84\cdot 101^{4} +O(101^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |