Basic invariants
Dimension: | $2$ |
Group: | $D_{10}$ |
Conductor: | \(4475\)\(\medspace = 5^{2} \cdot 179 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.0.574268740309375.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{10}$ |
Parity: | odd |
Determinant: | 1.179.2t1.a.a |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.32041.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 2x^{9} + 7x^{8} + 20x^{7} - 18x^{6} + 146x^{5} + 99x^{4} + 188x^{3} + 1095x^{2} - 1376x + 3136 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{5} + 5x + 17 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a^{4} + 14 a^{2} + 15 a + 8 + \left(15 a^{4} + 13 a^{3} + 11 a^{2} + 15 a + 18\right)\cdot 19 + \left(12 a^{4} + 15 a^{3} + 2 a^{2} + 16\right)\cdot 19^{2} + \left(7 a^{4} + 16 a^{3} + 18 a^{2} + 8 a + 7\right)\cdot 19^{3} + \left(a^{4} + 18 a^{3} + 6 a^{2} + 15 a + 9\right)\cdot 19^{4} + \left(18 a^{4} + 11 a^{3} + 12 a^{2} + 9 a + 11\right)\cdot 19^{5} + \left(3 a^{4} + 13 a^{3} + 5 a^{2} + 6 a\right)\cdot 19^{6} + \left(15 a^{4} + 8 a^{3} + 12 a^{2} + 2 a\right)\cdot 19^{7} + \left(5 a^{4} + 10 a^{3} + 12 a^{2} + 18 a + 8\right)\cdot 19^{8} + \left(9 a^{4} + 16 a^{3} + 9 a^{2} + 11 a + 14\right)\cdot 19^{9} +O(19^{10})\)
$r_{ 2 }$ |
$=$ |
\( 3 a^{4} + 12 a^{3} + 6 a^{2} + 8 a + 16 + \left(5 a^{4} + 6 a^{3} + 16 a^{2} + 16\right)\cdot 19 + \left(7 a^{4} + 3 a^{3} + 3 a^{2} + 11 a + 13\right)\cdot 19^{2} + \left(5 a^{4} + 3 a^{3} + a + 17\right)\cdot 19^{3} + \left(10 a^{4} + a^{3} + 7 a^{2} + 3 a + 6\right)\cdot 19^{4} + \left(7 a^{4} + 13 a^{3} + 5 a^{2} + 13 a + 7\right)\cdot 19^{5} + \left(9 a^{4} + 2 a^{3} + 9 a^{2} + 14 a + 3\right)\cdot 19^{6} + \left(2 a^{4} + 2 a^{3} + 5 a^{2} + a + 6\right)\cdot 19^{7} + \left(3 a^{4} + 13 a^{3} + 7 a^{2} + 2 a + 16\right)\cdot 19^{8} + \left(14 a^{4} + 18 a^{3} + 2 a^{2} + 4 a + 14\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 3 a^{4} + 18 a^{3} + 12 a^{2} + a + 16 + \left(15 a^{4} + 17 a^{3} + 12 a^{2} + 8 a + 18\right)\cdot 19 + \left(18 a^{4} + 12 a^{3} + 8 a^{2} + a + 2\right)\cdot 19^{2} + \left(2 a^{4} + 5 a^{3} + 12 a + 8\right)\cdot 19^{3} + \left(7 a^{4} + 2 a^{3} + 16 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(11 a^{4} + 11 a^{3} + 8 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(8 a^{4} + 13 a^{3} + 2 a^{2} + 9 a\right)\cdot 19^{6} + \left(8 a^{3} + 12 a^{2} + 14 a + 17\right)\cdot 19^{7} + \left(13 a^{4} + 12 a^{3} + 11 a^{2} + 17 a + 17\right)\cdot 19^{8} + \left(8 a^{4} + 4 a^{3} + 13 a^{2} + 5 a + 11\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a^{4} + 2 a^{3} + 4 a^{2} + 10 a + 9 + \left(a^{4} + 6 a^{3} + 2 a + 1\right)\cdot 19 + \left(10 a^{3} + 9 a^{2} + 4\right)\cdot 19^{2} + \left(5 a^{4} + 10 a^{3} + 12 a + 16\right)\cdot 19^{3} + \left(12 a^{4} + 15 a^{3} + 3 a^{2} + 17 a + 14\right)\cdot 19^{4} + \left(15 a^{4} + 9 a^{3} + 10 a^{2} + 13 a + 1\right)\cdot 19^{5} + \left(6 a^{4} + 18 a^{3} + 7 a^{2} + 10 a + 12\right)\cdot 19^{6} + \left(8 a^{4} + 3 a^{2} + a + 10\right)\cdot 19^{7} + \left(3 a^{3} + 10 a^{2} + 16 a + 5\right)\cdot 19^{8} + \left(13 a^{4} + 16 a^{3} + 3 a^{2} + 2 a + 10\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a^{4} + 9 a^{3} + 13 a^{2} + 6 + \left(5 a^{4} + 18 a^{3} + a^{2} + 18 a + 18\right)\cdot 19 + \left(12 a^{4} + 6 a^{3} + 15 a^{2} + 10 a + 14\right)\cdot 19^{2} + \left(5 a^{4} + 3 a^{3} + a^{2} + 4 a + 18\right)\cdot 19^{3} + \left(14 a^{4} + 18 a^{3} + 9 a^{2} + 6 a + 3\right)\cdot 19^{4} + \left(a^{4} + 2 a^{3} + 4 a^{2} + 3\right)\cdot 19^{5} + \left(8 a^{4} + 15 a^{3} + 13 a^{2} + 2 a + 17\right)\cdot 19^{6} + \left(a^{4} + 3 a^{3} + 15 a^{2} + 3 a + 1\right)\cdot 19^{7} + \left(2 a^{4} + 9 a^{3} + 2 a^{2} + 12\right)\cdot 19^{8} + \left(4 a^{4} + 18 a^{3} + a + 12\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 11 a^{4} + 4 a^{3} + 4 a^{2} + a + 10 + \left(2 a^{4} + 11 a^{3} + 16 a^{2} + 2 a + 6\right)\cdot 19 + \left(6 a^{4} + 3 a^{3} + 8 a^{2} + 2 a + 9\right)\cdot 19^{2} + \left(2 a^{4} + 17 a^{3} + 6 a^{2} + 10 a + 5\right)\cdot 19^{3} + \left(4 a^{4} + 10 a^{3} + 14 a + 1\right)\cdot 19^{4} + \left(4 a^{4} + 13 a^{3} + 11 a^{2} + 15 a + 13\right)\cdot 19^{5} + \left(8 a^{4} + 7 a^{3} + 7 a^{2} + 17 a + 17\right)\cdot 19^{6} + \left(5 a^{4} + a^{3} + 17 a^{2} + 10 a + 17\right)\cdot 19^{7} + \left(16 a^{4} + 15 a^{3} + 10 a^{2} + 16 a + 11\right)\cdot 19^{8} + \left(6 a^{3} + 14 a^{2} + 12 a + 18\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 11 a^{4} + 4 a^{3} + 16 a^{2} + 8 a + 10 + \left(9 a^{4} + 7 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 19 + \left(11 a^{4} + 13 a^{3} + 13 a^{2} + 6 a + 11\right)\cdot 19^{2} + \left(9 a^{4} + 12 a^{3} + 15 a^{2} + 15\right)\cdot 19^{3} + \left(2 a^{4} + 9 a^{3} + 7 a^{2} + 17 a + 13\right)\cdot 19^{4} + \left(13 a^{4} + a^{3} + 8 a^{2} + 6 a + 10\right)\cdot 19^{5} + \left(a^{4} + 13 a^{3} + 4 a^{2} + 16 a + 10\right)\cdot 19^{6} + \left(15 a^{4} + 5 a^{3} + 17 a^{2} + 3 a + 18\right)\cdot 19^{7} + \left(2 a^{4} + 17 a^{3} + 13 a^{2} + 14\right)\cdot 19^{8} + \left(4 a^{4} + a^{3} + 5 a^{2} + 17 a + 12\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 15 a^{4} + 5 a^{3} + a^{2} + 17 a + 7 + \left(6 a^{4} + 15 a^{3} + 16 a^{2} + 5 a + 4\right)\cdot 19 + \left(17 a^{4} + 9 a^{3} + 7 a^{2} + 18 a + 16\right)\cdot 19^{2} + \left(8 a^{4} + 2 a^{3} + 18 a^{2} + 5 a + 12\right)\cdot 19^{3} + \left(17 a^{4} + 8 a^{3} + 17 a^{2} + 3 a + 16\right)\cdot 19^{4} + \left(14 a^{3} + 16 a^{2} + 3 a + 18\right)\cdot 19^{5} + \left(6 a^{4} + 5 a^{3} + 11 a^{2} + 5 a + 8\right)\cdot 19^{6} + \left(5 a^{4} + 2 a^{3} + 8 a^{2} + 9 a + 17\right)\cdot 19^{7} + \left(17 a^{4} + 10 a^{3} + 6 a^{2} + 6 a + 15\right)\cdot 19^{8} + \left(16 a^{4} + 5 a^{3} + 6 a^{2} + 18 a + 6\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 17 a^{4} + a^{3} + 17 a^{2} + 16 a + 15 + \left(6 a^{4} + 5 a^{3} + 14 a^{2} + 7 a + 4\right)\cdot 19 + \left(6 a^{4} + 14 a^{3} + 8 a^{2} + 10\right)\cdot 19^{2} + \left(10 a^{4} + 13 a^{3} + 3 a^{2} + 16 a + 18\right)\cdot 19^{3} + \left(11 a^{4} + 11 a^{3} + 13 a^{2} + 3 a + 11\right)\cdot 19^{4} + \left(2 a^{4} + a^{3} + a^{2} + 13 a + 6\right)\cdot 19^{5} + \left(16 a^{4} + 9 a^{3} + 11 a^{2} + 8 a + 11\right)\cdot 19^{6} + \left(15 a^{4} + a^{3} + 18 a^{2} + 9 a + 2\right)\cdot 19^{7} + \left(6 a^{4} + 13 a^{3} + 12 a^{2} + a + 12\right)\cdot 19^{8} + \left(16 a^{4} + 3 a^{3} + 16 a^{2} + 2 a + 4\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 10 }$ |
$=$ |
\( 18 a^{4} + 2 a^{3} + 8 a^{2} + \left(7 a^{4} + 13 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 19 + \left(2 a^{4} + 4 a^{3} + 16 a^{2} + 5 a + 13\right)\cdot 19^{2} + \left(18 a^{4} + 9 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 19^{3} + \left(13 a^{4} + 17 a^{3} + 13 a^{2} + a + 2\right)\cdot 19^{4} + \left(14 a^{3} + 15 a^{2} + 15 a + 18\right)\cdot 19^{5} + \left(7 a^{4} + 14 a^{3} + 2 a^{2} + 3 a + 12\right)\cdot 19^{6} + \left(6 a^{4} + 2 a^{3} + 3 a^{2} + 2\right)\cdot 19^{7} + \left(8 a^{4} + 10 a^{3} + 6 a^{2} + 16 a + 18\right)\cdot 19^{8} + \left(7 a^{4} + 2 a^{3} + 3 a^{2} + 18 a + 6\right)\cdot 19^{9} +O(19^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,8)(3,9)(4,10)(5,7)$ | $-2$ |
$5$ | $2$ | $(1,9)(2,4)(3,6)(8,10)$ | $0$ |
$5$ | $2$ | $(1,3)(2,10)(4,8)(5,7)(6,9)$ | $0$ |
$2$ | $5$ | $(1,2,7,4,9)(3,6,8,5,10)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,7,9,2,4)(3,8,10,6,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $10$ | $(1,8,7,10,9,6,2,5,4,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$2$ | $10$ | $(1,10,2,3,7,6,4,8,9,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.