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 4.0.3519605.1 |
| 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
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 10x^{2} - 9x + 230 \)
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The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 97 + 124\cdot 211 + 56\cdot 211^{2} + 150\cdot 211^{3} + 128\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 102 + 181\cdot 211 + 151\cdot 211^{2} + 161\cdot 211^{3} + 144\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 110 + 29\cdot 211 + 59\cdot 211^{2} + 49\cdot 211^{3} + 66\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 115 + 86\cdot 211 + 154\cdot 211^{2} + 60\cdot 211^{3} + 82\cdot 211^{4} +O(211^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |