Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(143\)\(\medspace = 11 \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.20449.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Determinant: | 1.143.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.20449.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + 3x - 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 13 a + 3 + \left(2 a + 11\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 9\right)\cdot 17^{3} + \left(13 a + 16\right)\cdot 17^{4} +O(17^{5})\)
$r_{ 2 }$ |
$=$ |
\( 9 a + 7 + \left(a + 8\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + \left(a + 16\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 + 12\cdot 17 + 14\cdot 17^{2} + 9\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 8 a + 16 + 15 a\cdot 17 + \left(8 a + 12\right)\cdot 17^{2} + \left(15 a + 9\right)\cdot 17^{3} + \left(5 a + 12\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 16 + 14 a\cdot 17 + \left(3 a + 6\right)\cdot 17^{2} + \left(8 a + 5\right)\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} +O(17^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,3)(2,5)$ | $0$ |
$2$ | $5$ | $(1,5,2,3,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,2,4,5,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.