Basic invariants
Dimension: | $2$ |
Group: | $D_{10}$ |
Conductor: | \(2159\)\(\medspace = 17 \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.0.2759397651242047.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{10}$ |
Parity: | odd |
Determinant: | 1.2159.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.4661281.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 5x^{9} + 23x^{8} - 40x^{7} + 29x^{6} + 85x^{5} + 58x^{4} - 160x^{3} + 184x^{2} - 80x + 32 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 5 a^{4} + a^{3} + 4 a^{2} + 7 a + \left(9 a^{4} + 5 a^{3} + 9 a^{2} + 10 a + 9\right)\cdot 13 + \left(4 a^{4} + 8 a^{3} + 7 a^{2} + 10 a\right)\cdot 13^{2} + \left(7 a^{4} + 4 a^{3} + a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(12 a^{4} + 2 a^{3} + 8 a^{2} + 5 a\right)\cdot 13^{4} + \left(9 a^{4} + 2 a^{3} + 6 a^{2} + 7 a + 9\right)\cdot 13^{5} + \left(2 a^{4} + 8 a^{3} + 2 a^{2} + 4\right)\cdot 13^{6} + \left(9 a^{4} + 3 a^{3} + 10 a^{2} + 6 a + 6\right)\cdot 13^{7} + \left(4 a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 3\right)\cdot 13^{8} + \left(4 a^{4} + 6 a^{3} + 10 a^{2} + 9 a + 7\right)\cdot 13^{9} +O(13^{10})\)
$r_{ 2 }$ |
$=$ |
\( 10 a^{4} + 11 a^{3} + 8 a^{2} + 10 + \left(2 a^{4} + 3 a^{3} + 3 a^{2} + 5 a + 11\right)\cdot 13 + \left(9 a^{3} + 3 a^{2} + 3 a + 9\right)\cdot 13^{2} + \left(10 a^{4} + 7 a^{3} + 6 a^{2} + 5 a + 7\right)\cdot 13^{3} + \left(12 a^{4} + 5 a^{3} + 5 a^{2} + 6 a + 5\right)\cdot 13^{4} + \left(3 a^{4} + 5 a^{3} + 9 a^{2} + 2 a + 4\right)\cdot 13^{5} + \left(10 a^{4} + 12 a^{3} + 8 a^{2} + 3 a + 3\right)\cdot 13^{6} + \left(2 a^{4} + 7 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13^{7} + \left(12 a^{4} + 2 a^{3} + 2 a^{2} + 4 a + 3\right)\cdot 13^{8} + \left(12 a^{4} + 12 a^{3} + 4 a^{2} + 8 a + 1\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 12 a^{3} + 10 a^{2} + 12 a + 4 + \left(11 a^{4} + 10 a^{3} + 3 a^{2} + 12 a + 4\right)\cdot 13 + \left(10 a^{4} + a^{3} + 11 a^{2} + 9 a + 2\right)\cdot 13^{2} + \left(11 a^{4} + a^{3} + 12 a^{2} + 11 a + 3\right)\cdot 13^{3} + \left(10 a^{4} + 2 a^{3} + 5 a^{2} + 3 a + 2\right)\cdot 13^{4} + \left(5 a^{4} + 8 a^{2} + 4 a + 5\right)\cdot 13^{5} + \left(6 a^{4} + 3 a^{3} + 12 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(10 a^{4} + 6 a^{2} + 11 a + 2\right)\cdot 13^{7} + \left(a^{4} + 11 a^{3} + a^{2} + 4 a + 12\right)\cdot 13^{8} + \left(11 a^{4} + 2 a^{3} + 4 a^{2} + 2 a + 5\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{4} + 5 a^{3} + 6 a^{2} + 9 + \left(8 a^{4} + 9 a^{3} + 12 a^{2} + 9 a + 3\right)\cdot 13 + \left(2 a^{4} + 12 a^{3} + 2 a^{2} + 9 a + 12\right)\cdot 13^{2} + \left(12 a^{4} + 4 a^{3} + 9 a^{2} + 5 a\right)\cdot 13^{3} + \left(11 a^{4} + 10 a^{3} + 8 a^{2} + 11 a + 11\right)\cdot 13^{4} + \left(5 a^{4} + 12 a^{3} + 9 a^{2} + 3 a + 3\right)\cdot 13^{5} + \left(8 a^{4} + 12 a^{3} + 9 a^{2} + 10 a + 7\right)\cdot 13^{6} + \left(9 a^{4} + 7 a^{2} + 7 a + 2\right)\cdot 13^{7} + \left(a^{4} + 9 a^{3} + 5 a^{2} + 7 a + 12\right)\cdot 13^{8} + \left(6 a^{4} + 9 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 5 }$ |
$=$ |
\( a^{4} + 10 a^{3} + 3 a^{2} + 10 a + 8 + \left(12 a^{4} + 7 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 13 + \left(a^{4} + 3 a^{3} + 12 a^{2} + 11 a + 7\right)\cdot 13^{2} + \left(3 a^{4} + 11 a^{3} + 6 a^{2} + 8 a\right)\cdot 13^{3} + \left(12 a^{3} + 4 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(9 a^{4} + 3 a^{2} + 4 a\right)\cdot 13^{5} + \left(9 a^{4} + 9 a^{3} + 10 a + 6\right)\cdot 13^{6} + \left(7 a^{4} + 5 a^{3} + 6 a^{2} + 10 a + 9\right)\cdot 13^{7} + \left(10 a^{4} + 11 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 13^{8} + \left(12 a^{4} + 5 a^{3} + 6 a^{2} + 2 a\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a^{4} + 6 a^{3} + 7 a^{2} + 2 a + 5 + \left(a^{4} + 4 a^{3} + 4 a^{2} + 7 a + 2\right)\cdot 13 + \left(11 a^{4} + 4 a^{3} + 4 a^{2} + 5 a + 3\right)\cdot 13^{2} + \left(10 a^{4} + 12 a^{3} + 12 a^{2} + 12 a + 5\right)\cdot 13^{3} + \left(3 a^{4} + 4 a^{3} + 8 a^{2} + 5 a\right)\cdot 13^{4} + \left(5 a^{4} + 6 a^{3} + 7 a^{2} + 5 a + 6\right)\cdot 13^{5} + \left(2 a^{4} + a^{2} + 2 a + 1\right)\cdot 13^{6} + \left(3 a^{4} + 2 a^{3} + a^{2} + a + 2\right)\cdot 13^{7} + \left(5 a^{4} + 10 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 13^{8} + \left(8 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 12\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 5 a^{4} + 2 a^{3} + 2 a^{2} + 12 a + 7 + \left(2 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 13 + \left(6 a^{4} + 12 a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 13^{2} + \left(12 a^{4} + 6 a^{3} + 3 a + 7\right)\cdot 13^{3} + \left(3 a^{4} + 6 a^{3} + 11 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(10 a^{4} + 10 a^{3} + 2 a + 11\right)\cdot 13^{5} + \left(2 a^{4} + a^{3} + 4 a^{2} + 7 a + 7\right)\cdot 13^{6} + \left(7 a^{4} + a^{3} + 10 a^{2} + 6 a + 4\right)\cdot 13^{7} + \left(2 a^{4} + 4 a^{3} + 6 a^{2} + a + 9\right)\cdot 13^{8} + \left(11 a^{3} + 6 a^{2} + 11 a + 4\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 2 a^{4} + 6 a^{3} + 4 a^{2} + 3 a + 6 + \left(2 a^{4} + 11 a^{3} + 5 a + 1\right)\cdot 13 + \left(6 a^{4} + 5 a^{3} + 11 a^{2}\right)\cdot 13^{2} + \left(a^{4} + 7 a^{3} + 6 a^{2} + 3 a + 3\right)\cdot 13^{3} + \left(12 a^{4} + 6 a^{2} + 10 a + 1\right)\cdot 13^{4} + \left(3 a^{4} + 6 a^{3} + 2 a^{2} + 11 a\right)\cdot 13^{5} + \left(5 a^{4} + 7 a^{2} + 6 a + 5\right)\cdot 13^{6} + \left(11 a^{4} + 8 a^{3} + 12 a^{2} + 10 a\right)\cdot 13^{7} + \left(a^{4} + 12 a^{3} + 2 a^{2} + 10\right)\cdot 13^{8} + \left(6 a^{4} + 12 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 11 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 1 + \left(6 a^{4} + 5 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 13 + \left(10 a^{4} + 8 a^{3} + 4 a^{2} + 6 a + 11\right)\cdot 13^{2} + \left(a^{4} + 10 a^{3} + a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(2 a^{4} + 12 a^{3} + 11 a^{2} + 4 a + 10\right)\cdot 13^{4} + \left(10 a^{4} + 3 a^{3} + 3 a^{2} + 11 a + 1\right)\cdot 13^{5} + \left(12 a^{4} + 8 a^{3} + 6 a^{2} + 10 a + 8\right)\cdot 13^{6} + \left(7 a^{3} + 2 a^{2} + 3 a + 8\right)\cdot 13^{7} + \left(7 a^{4} + 12 a^{3} + a^{2} + a\right)\cdot 13^{8} + \left(9 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 3\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 10 }$ |
$=$ |
\( 5 a^{4} + 8 a^{3} + 12 a^{2} + 7 + \left(8 a^{4} + 4 a^{3} + 7 a^{2} + 8 a + 11\right)\cdot 13 + \left(10 a^{4} + 11 a^{3} + 4 a^{2} + 11 a + 6\right)\cdot 13^{2} + \left(6 a^{4} + 10 a^{3} + 7 a^{2} + 5 a + 10\right)\cdot 13^{3} + \left(7 a^{4} + 6 a^{3} + 7 a^{2} + a + 1\right)\cdot 13^{4} + \left(3 a^{3} + 12 a^{2} + 11 a + 9\right)\cdot 13^{5} + \left(4 a^{4} + 8 a^{3} + 11 a^{2} + 4 a + 6\right)\cdot 13^{6} + \left(2 a^{4} + a^{3} + 12 a^{2} + 3 a + 4\right)\cdot 13^{7} + \left(4 a^{4} + 11 a^{3} + 8 a^{2} + 7 a + 9\right)\cdot 13^{8} + \left(6 a^{4} + 8 a^{3} + 12 a^{2} + 7 a + 8\right)\cdot 13^{9} +O(13^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,10)(2,9)(3,8)(4,7)(5,6)$ | $-2$ |
$5$ | $2$ | $(1,3)(2,9)(4,6)(5,7)(8,10)$ | $0$ |
$5$ | $2$ | $(1,8)(3,10)(4,5)(6,7)$ | $0$ |
$2$ | $5$ | $(1,9,8,5,4)(2,3,6,7,10)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,5,9,4,8)(2,7,3,10,6)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,6,9,7,8,10,5,2,4,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,7,5,3,9,10,4,6,8,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.