Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(4195\)\(\medspace = 5 \cdot 839 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.20975.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4195.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-839})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 4x^{2} + 15x - 55 \)
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The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 15\cdot 211 + 96\cdot 211^{2} + 42\cdot 211^{3} + 112\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 131\cdot 211 + 98\cdot 211^{2} + 152\cdot 211^{3} + 49\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 + 109\cdot 211 + 171\cdot 211^{2} + 107\cdot 211^{3} + 16\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 92 + 166\cdot 211 + 55\cdot 211^{2} + 119\cdot 211^{3} + 32\cdot 211^{4} +O(211^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ | |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ | |
| $2$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |