Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(131\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 5.1.17161.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | odd |
Projective image: | $D_5$ |
Projective field: | Galois closure of 5.1.17161.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ | \( 15 + 10\cdot 17 + 15\cdot 17^{2} + 13\cdot 17^{3} + 4\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 a + 1 + \left(9 a + 12\right)\cdot 17 + \left(15 a + 2\right)\cdot 17^{2} + \left(15 a + 11\right)\cdot 17^{3} + \left(14 a + 1\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 5 a + 13 + \left(7 a + 9\right)\cdot 17 + \left(a + 8\right)\cdot 17^{2} + \left(a + 11\right)\cdot 17^{3} + 2 a\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 a + \left(12 a + 6\right)\cdot 17 + a\cdot 17^{2} + \left(13 a + 10\right)\cdot 17^{3} + \left(14 a + 12\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 11 a + 6 + \left(4 a + 12\right)\cdot 17 + \left(15 a + 6\right)\cdot 17^{2} + \left(3 a + 4\right)\cdot 17^{3} + \left(2 a + 14\right)\cdot 17^{4} +O(17^{5})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$5$ | $2$ | $(1,2)(3,4)$ | $0$ | $0$ |
$2$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,3,5,4,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |