Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(127\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.16129.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Determinant: | 1.127.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.16129.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 2x^{3} + x^{2} + 3x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a + 3 + \left(5 a + 6\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(a + 6\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 2 }$ | $=$ | \( a + 2 + \left(a + 6\right)\cdot 7 + \left(4 a + 5\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 3 }$ | $=$ | \( 5 + 3\cdot 7 + 5\cdot 7^{3} + 7^{4} +O(7^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 a + \left(4 a + 6\right)\cdot 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 2 a\cdot 7^{4} +O(7^{5})\) |
$r_{ 5 }$ | $=$ | \( 2 a + 5 + \left(2 a + 5\right)\cdot 7 + 6 a\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} +O(7^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,3)(2,4)$ | $0$ |
$2$ | $5$ | $(1,2,4,3,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,4,5,2,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.