Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(847\)\(\medspace = 7 \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.717409.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.717409.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + 6x^{3} - 9x^{2} + 4x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 7 a + 11 + \left(12 a + 7\right)\cdot 13 + \left(9 a + 10\right)\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + \left(5 a + 12\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 4 a + \left(11 a + 3\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + 12\cdot 13^{3} + \left(10 a + 10\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 a + 4 + \left(a + 10\right)\cdot 13 + \left(5 a + 6\right)\cdot 13^{2} + \left(12 a + 5\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 + 4\cdot 13 + 3\cdot 13^{2} + 12\cdot 13^{3} +O(13^{5})\) |
$r_{ 5 }$ | $=$ | \( 6 a + 5 + \left(3 a + 8\right)\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + \left(7 a + 7\right)\cdot 13^{4} +O(13^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,4)(2,5)$ | $0$ |
$2$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,3,4,2,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.