Basic invariants
Dimension: | $2$ |
Group: | $D_{10}$ |
Conductor: | \(20727\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 47 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.0.936668172433707.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{10}$ |
Parity: | odd |
Determinant: | 1.47.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.2209.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 4x^{9} + 16x^{8} + 8x^{7} + 135x^{6} + 98x^{5} + 770x^{4} + 4000x^{3} + 16523x^{2} - 55117x + 235531 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{5} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( 7 a^{4} + 16 a^{3} + 16 a + 6 + \left(8 a^{4} + 15 a^{3} + 15 a^{2} + 10 a + 3\right)\cdot 17 + \left(9 a^{4} + 10 a^{3} + 8 a^{2} + 8 a + 4\right)\cdot 17^{2} + \left(14 a^{4} + 2 a^{3} + 7 a^{2} + 7 a + 8\right)\cdot 17^{3} + \left(15 a^{4} + 13 a^{3} + 15 a + 2\right)\cdot 17^{4} + \left(15 a^{4} + 14 a^{3} + 14 a^{2} + 15 a + 16\right)\cdot 17^{5} + \left(9 a^{4} + 9 a^{3} + 12 a^{2} + 6 a + 7\right)\cdot 17^{6} + \left(4 a^{4} + 12 a^{3} + 15 a^{2} + 15 a + 10\right)\cdot 17^{7} + \left(10 a^{4} + 12 a^{2} + 11 a + 11\right)\cdot 17^{8} + \left(14 a^{4} + 9 a^{3} + 8 a^{2} + 13 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 2 }$ | $=$ | \( 8 a^{4} + 10 a^{3} + 7 a^{2} + 6 a + \left(3 a^{4} + 10 a^{3} + 8 a^{2} + 13 a + 13\right)\cdot 17 + \left(12 a^{4} + 15 a^{3} + 2 a^{2} + 9\right)\cdot 17^{2} + \left(3 a^{4} + 12 a^{3} + 5 a^{2} + 4 a + 16\right)\cdot 17^{3} + \left(a^{4} + 12 a^{3} + 11 a^{2} + a\right)\cdot 17^{4} + \left(a^{3} + a^{2} + 16 a\right)\cdot 17^{5} + \left(11 a^{4} + 8 a^{3} + 8 a^{2} + 7 a + 2\right)\cdot 17^{6} + \left(7 a^{4} + 6 a^{3} + 14 a^{2} + 16 a + 6\right)\cdot 17^{7} + \left(14 a^{4} + 10 a^{2} + 3 a + 1\right)\cdot 17^{8} + \left(15 a^{4} + 7 a^{3} + 2 a^{2} + 6 a + 16\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{4} + 8 a^{3} + 10 a^{2} + 7 a + 11 + \left(14 a^{4} + 12 a^{3} + 7 a^{2} + 2 a + 11\right)\cdot 17 + \left(11 a^{4} + 15 a^{3} + 6 a^{2} + 16 a + 2\right)\cdot 17^{2} + \left(a^{4} + 13 a^{3} + 3 a^{2} + 14 a + 8\right)\cdot 17^{3} + \left(14 a^{4} + 8 a^{3} + 4 a^{2} + 6 a + 4\right)\cdot 17^{4} + \left(10 a^{4} + 10 a^{3} + 8 a^{2} + a + 5\right)\cdot 17^{5} + \left(7 a^{4} + 3 a^{3} + 3 a^{2} + a + 16\right)\cdot 17^{6} + \left(10 a^{4} + 3 a^{3} + 10 a^{2} + 6 a + 4\right)\cdot 17^{7} + \left(7 a^{4} + 15 a^{3} + 4 a^{2} + 6 a + 16\right)\cdot 17^{8} + \left(2 a^{4} + 8 a^{3} + a^{2} + a + 1\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{4} + 10 a^{3} + 12 a^{2} + a + 16 + \left(4 a^{4} + 2 a^{3} + 10 a^{2} + 9 a + 13\right)\cdot 17 + \left(16 a^{4} + 16 a^{3} + a^{2} + 11 a + 2\right)\cdot 17^{2} + \left(13 a^{4} + 12 a^{3} + 13 a^{2} + 7 a + 11\right)\cdot 17^{3} + \left(8 a^{4} + 11 a^{3} + 8 a^{2} + 5 a + 3\right)\cdot 17^{4} + \left(11 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 9\right)\cdot 17^{5} + \left(16 a^{4} + 12 a^{3} + 10 a^{2} + 13 a + 6\right)\cdot 17^{6} + \left(14 a^{4} + 16 a^{3} + a^{2} + 15 a + 15\right)\cdot 17^{7} + \left(13 a^{4} + 14 a^{3} + 11 a^{2} + 2 a\right)\cdot 17^{8} + \left(16 a^{4} + 11 a^{3} + 5 a^{2} + a + 10\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 5 }$ | $=$ | \( 11 a^{4} + 10 a^{3} + 16 a^{2} + 12 a + 16 + \left(14 a^{4} + 3 a^{3} + 13 a^{2} + 16 a + 4\right)\cdot 17 + \left(12 a^{4} + 12 a^{3} + 9 a^{2} + 10 a + 10\right)\cdot 17^{2} + \left(2 a^{4} + 12 a^{3} + 16 a^{2} + 3 a + 5\right)\cdot 17^{3} + \left(12 a^{4} + 6 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 17^{4} + \left(13 a^{4} + 12 a^{3} + 13 a^{2} + 7 a + 14\right)\cdot 17^{5} + \left(7 a^{3} + a^{2} + 4 a + 10\right)\cdot 17^{6} + \left(8 a^{4} + 2 a^{3} + 4 a^{2} + 7 a + 16\right)\cdot 17^{7} + \left(5 a^{4} + 9 a^{3} + 14 a^{2} + 15 a\right)\cdot 17^{8} + \left(a^{4} + 7 a^{3} + a^{2} + a + 1\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 6 }$ | $=$ | \( 13 a^{4} + 7 a^{3} + 13 a^{2} + 15 a + 4 + \left(8 a^{4} + 16 a^{3} + 3 a^{2} + 11 a\right)\cdot 17 + \left(12 a^{3} + 7 a^{2} + 3 a + 14\right)\cdot 17^{2} + \left(a^{4} + 8 a^{3} + 10 a^{2}\right)\cdot 17^{3} + \left(10 a^{3} + 14 a^{2} + 3 a\right)\cdot 17^{4} + \left(16 a^{4} + a^{3} + 3 a^{2} + 15 a + 6\right)\cdot 17^{5} + \left(15 a^{4} + 5 a^{2} + 7 a + 9\right)\cdot 17^{6} + \left(12 a^{4} + 16 a^{3} + 2 a^{2} + 6 a + 3\right)\cdot 17^{7} + \left(13 a^{4} + 10 a^{3} + 8 a^{2} + 14 a + 4\right)\cdot 17^{8} + \left(15 a^{4} + 13 a^{3} + 16 a^{2} + 15 a + 9\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{4} + 9 a^{3} + 3 a^{2} + 16 a + 15 + \left(7 a^{4} + 7 a^{3} + 16 a^{2} + 11 a + 2\right)\cdot 17 + \left(12 a^{4} + a^{3} + a^{2} + 2 a + 3\right)\cdot 17^{2} + \left(16 a^{4} + 3 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 17^{3} + \left(5 a^{4} + 16 a^{3} + 5 a^{2} + 13 a + 8\right)\cdot 17^{4} + \left(6 a^{4} + 16 a^{3} + 6 a^{2} + a + 8\right)\cdot 17^{5} + \left(5 a^{4} + 10 a^{3} + 3 a^{2} + 2 a + 14\right)\cdot 17^{6} + \left(13 a^{4} + 8 a^{3} + 5 a^{2} + 14 a + 3\right)\cdot 17^{7} + \left(12 a^{4} + 13 a^{3} + 3 a^{2} + 14 a + 10\right)\cdot 17^{8} + \left(8 a^{4} + 9 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 8 }$ | $=$ | \( 15 a^{4} + 8 a^{3} + 3 a^{2} + 10 a + 9 + \left(6 a^{4} + 5 a^{3} + 12 a^{2} + 2 a + 5\right)\cdot 17 + \left(a^{4} + 13 a^{3} + 14 a + 11\right)\cdot 17^{2} + \left(5 a^{4} + 8 a^{3} + 6 a^{2} + 2 a\right)\cdot 17^{3} + \left(a^{4} + 11 a^{3} + 2 a^{2} + 2 a + 1\right)\cdot 17^{4} + \left(10 a^{4} + 7 a^{2} + 4 a + 8\right)\cdot 17^{5} + \left(6 a^{4} + 7 a^{3} + 4 a^{2} + 16 a + 15\right)\cdot 17^{6} + \left(5 a^{4} + 14 a^{3} + 9 a^{2} + 15 a\right)\cdot 17^{7} + \left(13 a^{4} + 9 a^{3} + 12 a^{2} + 14\right)\cdot 17^{8} + \left(11 a^{4} + 12 a^{3} + 8 a^{2} + 7 a + 12\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 9 }$ | $=$ | \( 15 a^{4} + 11 a^{3} + 14 a^{2} + 9 a + 9 + \left(7 a^{4} + 4 a^{3} + 3 a^{2} + 2 a + 16\right)\cdot 17 + \left(10 a^{4} + 11 a^{3} + 16 a^{2} + 5 a + 4\right)\cdot 17^{2} + \left(12 a^{4} + 8 a^{3} + 16 a^{2} + 16 a + 3\right)\cdot 17^{3} + \left(a^{4} + 15 a^{3} + 10 a + 8\right)\cdot 17^{4} + \left(2 a^{4} + 10 a^{3} + 14 a^{2} + 11 a + 8\right)\cdot 17^{5} + \left(9 a^{4} + a^{3} + 11 a^{2} + 4 a\right)\cdot 17^{6} + \left(10 a^{4} + 10 a^{3} + 16 a^{2} + 4 a + 5\right)\cdot 17^{7} + \left(7 a^{4} + 2 a^{3} + 4 a^{2} + 6 a + 16\right)\cdot 17^{8} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 10 }$ | $=$ | \( 16 a^{4} + 13 a^{3} + 7 a^{2} + 10 a + 3 + \left(7 a^{4} + 5 a^{3} + 10 a^{2} + 3 a + 13\right)\cdot 17 + \left(14 a^{4} + 9 a^{3} + 12 a^{2} + 11 a + 4\right)\cdot 17^{2} + \left(12 a^{4} + 8 a^{2} + 10\right)\cdot 17^{3} + \left(6 a^{4} + 12 a^{3} + 13 a^{2} + 6 a + 15\right)\cdot 17^{4} + \left(15 a^{4} + 3 a^{3} + 4 a^{2} + 8\right)\cdot 17^{5} + \left(a^{4} + 6 a^{3} + 6 a^{2} + 3 a + 1\right)\cdot 17^{6} + \left(14 a^{4} + 11 a^{3} + 5 a^{2} + 1\right)\cdot 17^{7} + \left(2 a^{4} + 7 a^{3} + 2 a^{2} + 8 a + 9\right)\cdot 17^{8} + \left(8 a^{4} + 11 a^{3} + 13 a^{2} + 8 a + 6\right)\cdot 17^{9} +O(17^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,5)(3,8)(4,10)(6,9)$ | $-2$ |
$5$ | $2$ | $(1,10)(2,3)(4,7)(5,8)(6,9)$ | $0$ |
$5$ | $2$ | $(1,4)(2,8)(3,5)(7,10)$ | $0$ |
$2$ | $5$ | $(1,5,6,3,4)(2,9,8,10,7)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,3,5,4,6)(2,10,9,7,8)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $10$ | $(1,8,5,10,6,7,3,2,4,9)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$2$ | $10$ | $(1,10,3,9,5,7,4,8,6,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.