Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(2159\)\(\medspace = 17 \cdot 127 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.4661281.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Determinant: | 1.2159.2t1.a.a |
| Projective image: | $D_5$ |
| Projective stem field: | Galois closure of 5.1.4661281.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 13x^{3} + 41x^{2} - 48x + 25 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + \left(8 a + 2\right)\cdot 11 + \left(2 a + 6\right)\cdot 11^{2} + \left(3 a + 3\right)\cdot 11^{3} + \left(10 a + 3\right)\cdot 11^{4} + \left(8 a + 2\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 5 + \left(2 a + 9\right)\cdot 11 + \left(8 a + 8\right)\cdot 11^{2} + \left(7 a + 2\right)\cdot 11^{3} + 8\cdot 11^{4} + \left(2 a + 5\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 3 + \left(9 a + 5\right)\cdot 11 + \left(9 a + 6\right)\cdot 11^{2} + 5 a\cdot 11^{3} + \left(2 a + 7\right)\cdot 11^{4} + 3\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 2\cdot 11 + 8\cdot 11^{2} + 3\cdot 11^{4} + 8\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 6 + \left(a + 2\right)\cdot 11 + \left(a + 3\right)\cdot 11^{2} + \left(5 a + 3\right)\cdot 11^{3} + 8 a\cdot 11^{4} + \left(10 a + 2\right)\cdot 11^{5} +O(11^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $5$ | $(1,5,3,2,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.