Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(475\)\(\medspace = 5^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.2375.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + x^{2} - 6x - 4 \)
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The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 + 98\cdot 131 + 38\cdot 131^{2} + 44\cdot 131^{3} + 68\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 84 + 106\cdot 131 + 47\cdot 131^{2} + 95\cdot 131^{3} + 15\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 123 + 15\cdot 131 + 53\cdot 131^{2} + 54\cdot 131^{3} + 38\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 128 + 40\cdot 131 + 122\cdot 131^{2} + 67\cdot 131^{3} + 8\cdot 131^{4} +O(131^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ | |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ | |
| $2$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |