Basic invariants
Dimension: | $2$ |
Group: | $D_{10}$ |
Conductor: | \(15296\)\(\medspace = 2^{6} \cdot 239 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.2.106915713548288.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{10}$ |
Parity: | odd |
Determinant: | 1.239.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.57121.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} + 8x^{8} - 40x^{4} + 224x^{2} - 32 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{5} + 7x + 28 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{4} + 16 a^{3} + 13 a^{2} + 9 a + 21 + \left(26 a^{4} + 29 a^{3} + 17 a^{2} + 2 a + 4\right)\cdot 31 + \left(4 a^{4} + 6 a^{3} + 15 a^{2} + 13 a + 1\right)\cdot 31^{2} + \left(27 a^{4} + 22 a^{3} + 27 a^{2} + 19 a + 29\right)\cdot 31^{3} + \left(12 a^{4} + 29 a^{3} + 6 a^{2} + 10 a + 6\right)\cdot 31^{4} + \left(19 a^{4} + 16 a^{3} + 25 a^{2} + 10 a + 12\right)\cdot 31^{5} + \left(19 a^{4} + 13 a^{3} + 27 a^{2} + 5 a + 7\right)\cdot 31^{6} + \left(10 a^{4} + 19 a^{3} + 3 a^{2} + 22 a + 26\right)\cdot 31^{7} + \left(13 a^{4} + 4 a^{3} + 6 a^{2} + 5 a + 27\right)\cdot 31^{8} + \left(7 a^{4} + 19 a^{3} + 27 a^{2} + 28 a + 13\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( a^{4} + 16 a^{3} + 26 a^{2} + 19 a + 21 + \left(a^{4} + 19 a^{3} + 17 a^{2} + 26 a + 19\right)\cdot 31 + \left(2 a^{3} + 25 a^{2} + 12 a + 17\right)\cdot 31^{2} + \left(15 a^{3} + 28 a^{2} + 2 a + 19\right)\cdot 31^{3} + \left(3 a^{4} + 29 a^{3} + 20 a^{2} + 8 a + 13\right)\cdot 31^{4} + \left(29 a^{4} + 4 a^{3} + 19 a^{2} + 5 a + 4\right)\cdot 31^{5} + \left(27 a^{4} + 3 a^{3} + 8 a^{2} + 28 a + 29\right)\cdot 31^{6} + \left(12 a^{4} + 3 a^{3} + 3 a^{2} + 22 a + 7\right)\cdot 31^{7} + \left(7 a^{4} + 30 a^{3} + 7 a^{2} + 20 a + 13\right)\cdot 31^{8} + \left(19 a^{4} + 20 a^{3} + 11 a^{2} + 11 a + 30\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{4} + 20 a^{3} + 4 a^{2} + 10 a + 2 + \left(8 a^{4} + 30 a^{3} + 28 a^{2} + 30 a + 25\right)\cdot 31 + \left(21 a^{4} + 3 a^{3} + 9 a^{2} + 20 a + 14\right)\cdot 31^{2} + \left(11 a^{4} + 28 a^{3} + 29 a^{2} + 25 a + 2\right)\cdot 31^{3} + \left(27 a^{3} + 12 a^{2} + 12 a + 30\right)\cdot 31^{4} + \left(17 a^{4} + 14 a^{3} + 28 a^{2} + 27 a + 11\right)\cdot 31^{5} + \left(12 a^{4} + 17 a^{3} + 27 a^{2} + 3 a + 5\right)\cdot 31^{6} + \left(20 a^{4} + 27 a^{3} + a^{2} + 21 a + 11\right)\cdot 31^{7} + \left(27 a^{4} + 17 a^{3} + 17 a^{2} + 20 a + 3\right)\cdot 31^{8} + \left(30 a^{4} + 11 a^{3} + 4 a^{2} + 14 a + 27\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{4} + 3 a^{3} + 10 a^{2} + 26 + \left(16 a^{4} + 17 a^{3} + 9 a^{2} + 5 a + 4\right)\cdot 31 + \left(16 a^{4} + 2 a^{3} + 6 a^{2} + 27 a + 23\right)\cdot 31^{2} + \left(5 a^{4} + 11 a^{3} + 21 a^{2} + 4 a + 25\right)\cdot 31^{3} + \left(6 a^{4} + 8 a^{3} + 26 a^{2} + 9 a + 12\right)\cdot 31^{4} + \left(21 a^{4} + 21 a^{3} + 26 a^{2} + 21 a + 28\right)\cdot 31^{5} + \left(2 a^{4} + 13 a^{3} + 21 a^{2} + 18 a + 17\right)\cdot 31^{6} + \left(19 a^{4} + 25 a^{3} + 6 a^{2} + 14 a + 11\right)\cdot 31^{7} + \left(3 a^{4} + 16 a^{3} + 8 a^{2} + 7 a + 4\right)\cdot 31^{8} + \left(3 a^{4} + 16 a^{3} + 12 a^{2} + 27 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 3 a^{4} + 15 a^{3} + 14 a^{2} + 18 a + 20 + \left(4 a^{4} + 4 a^{3} + 16 a^{2} + 3 a + 27\right)\cdot 31 + \left(8 a^{3} + 6 a^{2} + a + 1\right)\cdot 31^{2} + \left(21 a^{4} + 20 a^{3} + 17 a^{2} + a + 5\right)\cdot 31^{3} + \left(21 a^{4} + 8 a^{3} + 10 a^{2} + 15 a + 19\right)\cdot 31^{4} + \left(21 a^{4} + 28 a^{3} + 12 a^{2} + 9 a + 19\right)\cdot 31^{5} + \left(6 a^{4} + 12 a^{3} + 30 a^{2} + 17 a + 3\right)\cdot 31^{6} + \left(22 a^{4} + 20 a^{3} + 11 a^{2} + 7 a + 15\right)\cdot 31^{7} + \left(27 a^{4} + 2 a^{3} + 4 a^{2} + 13 a + 28\right)\cdot 31^{8} + \left(29 a^{4} + 14 a^{3} + 15 a^{2} + 21 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 28 a^{4} + 16 a^{3} + 17 a^{2} + 13 a + 11 + \left(26 a^{4} + 26 a^{3} + 14 a^{2} + 27 a + 3\right)\cdot 31 + \left(30 a^{4} + 22 a^{3} + 24 a^{2} + 29 a + 29\right)\cdot 31^{2} + \left(9 a^{4} + 10 a^{3} + 13 a^{2} + 29 a + 25\right)\cdot 31^{3} + \left(9 a^{4} + 22 a^{3} + 20 a^{2} + 15 a + 11\right)\cdot 31^{4} + \left(9 a^{4} + 2 a^{3} + 18 a^{2} + 21 a + 11\right)\cdot 31^{5} + \left(24 a^{4} + 18 a^{3} + 13 a + 27\right)\cdot 31^{6} + \left(8 a^{4} + 10 a^{3} + 19 a^{2} + 23 a + 15\right)\cdot 31^{7} + \left(3 a^{4} + 28 a^{3} + 26 a^{2} + 17 a + 2\right)\cdot 31^{8} + \left(a^{4} + 16 a^{3} + 15 a^{2} + 9 a + 28\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 28 a^{4} + 28 a^{3} + 21 a^{2} + 5 + \left(14 a^{4} + 13 a^{3} + 21 a^{2} + 26 a + 26\right)\cdot 31 + \left(14 a^{4} + 28 a^{3} + 24 a^{2} + 3 a + 7\right)\cdot 31^{2} + \left(25 a^{4} + 19 a^{3} + 9 a^{2} + 26 a + 5\right)\cdot 31^{3} + \left(24 a^{4} + 22 a^{3} + 4 a^{2} + 21 a + 18\right)\cdot 31^{4} + \left(9 a^{4} + 9 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 31^{5} + \left(28 a^{4} + 17 a^{3} + 9 a^{2} + 12 a + 13\right)\cdot 31^{6} + \left(11 a^{4} + 5 a^{3} + 24 a^{2} + 16 a + 19\right)\cdot 31^{7} + \left(27 a^{4} + 14 a^{3} + 22 a^{2} + 23 a + 26\right)\cdot 31^{8} + \left(27 a^{4} + 14 a^{3} + 18 a^{2} + 3 a + 28\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 29 a^{4} + 11 a^{3} + 27 a^{2} + 21 a + 29 + \left(22 a^{4} + 2 a^{2} + 5\right)\cdot 31 + \left(9 a^{4} + 27 a^{3} + 21 a^{2} + 10 a + 16\right)\cdot 31^{2} + \left(19 a^{4} + 2 a^{3} + a^{2} + 5 a + 28\right)\cdot 31^{3} + \left(30 a^{4} + 3 a^{3} + 18 a^{2} + 18 a\right)\cdot 31^{4} + \left(13 a^{4} + 16 a^{3} + 2 a^{2} + 3 a + 19\right)\cdot 31^{5} + \left(18 a^{4} + 13 a^{3} + 3 a^{2} + 27 a + 25\right)\cdot 31^{6} + \left(10 a^{4} + 3 a^{3} + 29 a^{2} + 9 a + 19\right)\cdot 31^{7} + \left(3 a^{4} + 13 a^{3} + 13 a^{2} + 10 a + 27\right)\cdot 31^{8} + \left(19 a^{3} + 26 a^{2} + 16 a + 3\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 30 a^{4} + 15 a^{3} + 5 a^{2} + 12 a + 10 + \left(29 a^{4} + 11 a^{3} + 13 a^{2} + 4 a + 11\right)\cdot 31 + \left(30 a^{4} + 28 a^{3} + 5 a^{2} + 18 a + 13\right)\cdot 31^{2} + \left(30 a^{4} + 15 a^{3} + 2 a^{2} + 28 a + 11\right)\cdot 31^{3} + \left(27 a^{4} + a^{3} + 10 a^{2} + 22 a + 17\right)\cdot 31^{4} + \left(a^{4} + 26 a^{3} + 11 a^{2} + 25 a + 26\right)\cdot 31^{5} + \left(3 a^{4} + 27 a^{3} + 22 a^{2} + 2 a + 1\right)\cdot 31^{6} + \left(18 a^{4} + 27 a^{3} + 27 a^{2} + 8 a + 23\right)\cdot 31^{7} + \left(23 a^{4} + 23 a^{2} + 10 a + 17\right)\cdot 31^{8} + \left(11 a^{4} + 10 a^{3} + 19 a^{2} + 19 a\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 10 }$ | $=$ | \( 30 a^{4} + 15 a^{3} + 18 a^{2} + 22 a + 10 + \left(4 a^{4} + a^{3} + 13 a^{2} + 28 a + 26\right)\cdot 31 + \left(26 a^{4} + 24 a^{3} + 15 a^{2} + 17 a + 29\right)\cdot 31^{2} + \left(3 a^{4} + 8 a^{3} + 3 a^{2} + 11 a + 1\right)\cdot 31^{3} + \left(18 a^{4} + a^{3} + 24 a^{2} + 20 a + 24\right)\cdot 31^{4} + \left(11 a^{4} + 14 a^{3} + 5 a^{2} + 20 a + 18\right)\cdot 31^{5} + \left(11 a^{4} + 17 a^{3} + 3 a^{2} + 25 a + 23\right)\cdot 31^{6} + \left(20 a^{4} + 11 a^{3} + 27 a^{2} + 8 a + 4\right)\cdot 31^{7} + \left(17 a^{4} + 26 a^{3} + 24 a^{2} + 25 a + 3\right)\cdot 31^{8} + \left(23 a^{4} + 11 a^{3} + 3 a^{2} + 2 a + 17\right)\cdot 31^{9} +O(31^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,10)(2,9)(3,8)(4,7)(5,6)$ | $-2$ |
$5$ | $2$ | $(1,9)(2,10)(3,4)(5,6)(7,8)$ | $0$ |
$5$ | $2$ | $(1,2)(3,7)(4,8)(9,10)$ | $0$ |
$2$ | $5$ | $(1,4,8,2,6)(3,9,5,10,7)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,8,6,4,2)(3,5,7,9,10)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,7,8,9,6,10,4,3,2,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$2$ | $10$ | $(1,9,4,5,8,10,2,7,6,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.