Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(295\)\(\medspace = 5 \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.1475.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.295.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-59})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - x^{2} + 5x - 5 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 52\cdot 79 + 39\cdot 79^{2} + 67\cdot 79^{3} + 43\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 51\cdot 79 + 68\cdot 79^{2} + 46\cdot 79^{3} + 78\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 + 61\cdot 79 + 62\cdot 79^{2} + 57\cdot 79^{3} + 26\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 76 + 71\cdot 79 + 65\cdot 79^{2} + 64\cdot 79^{3} + 8\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,3)$ | $0$ |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.