Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(159\)\(\medspace = 3 \cdot 53 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.25281.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Determinant: | 1.159.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.25281.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + x^{3} - 3x^{2} + x - 3 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 a + 9 + \left(9 a + 8\right)\cdot 11 + \left(2 a + 6\right)\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 9 a + 6 + \left(10 a + 2\right)\cdot 11 + \left(10 a + 3\right)\cdot 11^{2} + \left(4 a + 2\right)\cdot 11^{3} + \left(9 a + 8\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 7 a + 3 + \left(a + 9\right)\cdot 11 + \left(8 a + 8\right)\cdot 11^{2} + \left(a + 5\right)\cdot 11^{3} + \left(10 a + 1\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 2 a + 9 + 3\cdot 11 + 3\cdot 11^{2} + 6 a\cdot 11^{3} + \left(a + 8\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 5 }$ | $=$ | \( 6 + 8\cdot 11 + 10\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,2)(4,5)$ | $0$ |
$2$ | $5$ | $(1,3,2,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.