Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(1000\)\(\medspace = 2^{3} \cdot 5^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.1000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Determinant: | 1.40.2t1.b.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.1000000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 5x - 12 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 11 a + 5 + \left(2 a + 9\right)\cdot 17 + \left(16 a + 8\right)\cdot 17^{2} + 2 a\cdot 17^{3} + \left(6 a + 12\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 15 + 9\cdot 17 + 16\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 a + 14 + 8\cdot 17 + \left(7 a + 6\right)\cdot 17^{2} + \left(14 a + 11\right)\cdot 17^{3} + \left(4 a + 3\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 13 a + 1 + \left(16 a + 5\right)\cdot 17 + \left(9 a + 13\right)\cdot 17^{2} + \left(2 a + 1\right)\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 6 a + 16 + 14 a\cdot 17 + 5\cdot 17^{2} + \left(14 a + 4\right)\cdot 17^{3} + \left(10 a + 15\right)\cdot 17^{4} +O(17^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,5)(3,4)$ | $0$ |
$2$ | $5$ | $(1,5,3,2,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.