Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(920\)\(\medspace = 2^{3} \cdot 5 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.846400.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Determinant: | 1.920.2t1.a.a |
| Projective image: | $D_5$ |
| Projective stem field: | Galois closure of 5.1.846400.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 2x^{3} + 4x^{2} + 9x - 10 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 27 + \left(27 a + 1\right)\cdot 29 + \left(16 a + 14\right)\cdot 29^{2} + \left(4 a + 27\right)\cdot 29^{3} + \left(11 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 11 + \left(a + 4\right)\cdot 29 + \left(12 a + 13\right)\cdot 29^{2} + \left(24 a + 4\right)\cdot 29^{3} + \left(17 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 26 + \left(23 a + 1\right)\cdot 29 + \left(7 a + 27\right)\cdot 29^{2} + \left(6 a + 14\right)\cdot 29^{3} + \left(10 a + 6\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 12 + \left(5 a + 27\right)\cdot 29 + \left(21 a + 13\right)\cdot 29^{2} + \left(22 a + 9\right)\cdot 29^{3} + \left(18 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 22\cdot 29 + 18\cdot 29^{2} + 29^{3} + 20\cdot 29^{4} +O(29^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $2$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,3,4,2,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.