Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(50303545\)\(\medspace = 5 \cdot 19^{2} \cdot 29 \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.1458802805.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.145.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 1034x^{2} - 1618x + 107017 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 51\cdot 59 + 30\cdot 59^{2} + 39\cdot 59^{3} + 3\cdot 59^{4} + 8\cdot 59^{5} + 51\cdot 59^{6} + 21\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 21\cdot 59 + 15\cdot 59^{2} + 46\cdot 59^{3} + 27\cdot 59^{4} + 20\cdot 59^{5} + 46\cdot 59^{6} + 59^{7} +O(59^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 + 23\cdot 59 + 57\cdot 59^{2} + 47\cdot 59^{3} + 21\cdot 59^{4} + 14\cdot 59^{5} + 40\cdot 59^{6} + 20\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 21\cdot 59 + 14\cdot 59^{2} + 43\cdot 59^{3} + 5\cdot 59^{4} + 16\cdot 59^{5} + 39\cdot 59^{6} + 14\cdot 59^{7} +O(59^{8})\)
|
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.