Basic invariants
Dimension: | $1$ |
Group: | $C_{10}$ |
Conductor: | \(55\)\(\medspace = 5 \cdot 11 \) |
Artin field: | Galois closure of 10.10.669871503125.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{10}$ |
Parity: | even |
Dirichlet character: | \(\chi_{55}(4,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 46x^{6} - 11x^{5} - 52x^{4} + 7x^{3} + 18x^{2} - 3x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{5} + 5x + 17 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 a^{4} + 11 a^{3} + 8 a^{2} + 17 a + 9 + \left(a^{4} + 14 a^{3} + 11 a + 12\right)\cdot 19 + \left(7 a^{4} + 6 a^{3} + 2 a + 3\right)\cdot 19^{2} + \left(16 a^{4} + 4 a^{3} + 6 a^{2} + 18 a + 15\right)\cdot 19^{3} + \left(5 a^{4} + 9 a^{3} + 3 a^{2} + 15 a + 9\right)\cdot 19^{4} +O(19^{5})\)
$r_{ 2 }$ |
$=$ |
\( 2 a^{4} + 14 a^{3} + 2 a^{2} + 17 a + 11 + \left(5 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 8\right)\cdot 19 + \left(a^{4} + a^{3} + 13 a^{2} + 12 a + 14\right)\cdot 19^{2} + \left(4 a^{4} + 6 a^{3} + 4 a^{2} + 15 a + 5\right)\cdot 19^{3} + \left(8 a^{4} + 8 a^{3} + 8 a^{2} + 16 a + 12\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 4 a^{4} + 15 a^{3} + 17 a^{2} + 4 a + 17 + \left(6 a^{4} + 11 a^{3} + 15 a^{2} + 2 a + 13\right)\cdot 19 + \left(6 a^{4} + 14 a^{3} + 9 a^{2} + 5 a\right)\cdot 19^{2} + \left(8 a^{4} + a^{3} + 3 a^{2} + 4 a + 2\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 12 a^{2} + 3 a + 15\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a^{4} + a^{3} + 2 a^{2} + 4 + \left(16 a^{4} + a^{3} + 16 a^{2} + 10 a + 15\right)\cdot 19 + \left(2 a^{4} + 3 a^{3} + 8 a^{2} + 1\right)\cdot 19^{2} + \left(18 a^{4} + 15 a^{3} + 5 a^{2} + 10 a + 5\right)\cdot 19^{3} + \left(2 a^{4} + 9 a^{3} + 16 a^{2} + 17 a + 10\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a^{4} + 14 a^{3} + 5 a^{2} + 13 a + 8 + \left(7 a^{4} + 8 a^{3} + 17 a^{2} + 12 a + 17\right)\cdot 19 + \left(5 a^{4} + 7 a^{3} + 4 a^{2} + 11 a + 11\right)\cdot 19^{2} + \left(7 a^{4} + 3 a^{3} + a^{2} + 17 a + 18\right)\cdot 19^{3} + \left(17 a^{4} + 5 a^{3} + 8 a^{2} + 15 a + 10\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a^{4} + 11 a^{3} + 7 a^{2} + 12 a + 10 + \left(3 a^{4} + 6 a^{3} + 5 a^{2} + 16 a + 2\right)\cdot 19 + \left(16 a^{3} + 9 a^{2} + 3 a + 14\right)\cdot 19^{2} + \left(8 a^{4} + 12 a^{3} + 14 a^{2} + 5 a\right)\cdot 19^{3} + \left(15 a^{4} + 3 a^{3} + 15 a^{2} + 10 a + 10\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 8 a^{4} + 13 a^{3} + 7 a^{2} + 14 + \left(6 a^{4} + 10 a^{3} + 13 a^{2} + 16 a + 14\right)\cdot 19 + \left(14 a^{4} + 7 a^{3} + 8 a^{2} + 14 a + 13\right)\cdot 19^{2} + \left(13 a^{4} + 11 a^{3} + 7 a^{2} + 9 a + 4\right)\cdot 19^{3} + \left(15 a^{4} + 10 a^{3} + 14 a + 11\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 12 a^{4} + 7 a^{3} + 13 a^{2} + 12 a + 13 + \left(7 a^{4} + 8 a^{3} + 5 a^{2} + 14 a + 18\right)\cdot 19 + \left(14 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 19^{2} + \left(3 a^{4} + a^{3} + 9 a^{2} + 6 a + 4\right)\cdot 19^{3} + \left(12 a^{4} + 18 a^{3} + 8 a^{2} + 14 a + 9\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 9 }$ |
$=$ |
\( 13 a^{4} + 2 a^{3} + 16 a^{2} + 15 a + 17 + \left(a^{4} + 16 a^{3} + 6 a^{2} + 2 a + 13\right)\cdot 19 + \left(14 a^{4} + 17 a^{3} + 7 a^{2} + 10 a + 8\right)\cdot 19^{2} + \left(4 a^{4} + 11 a^{3} + 17 a^{2} + 7 a + 8\right)\cdot 19^{3} + \left(16 a^{4} + 15 a^{3} + 15 a^{2} + 11 a + 6\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 10 }$ |
$=$ |
\( 17 a^{4} + 7 a^{3} + 18 a^{2} + 5 a + 12 + \left(a^{4} + 13 a^{3} + 2 a^{2} + 10 a + 15\right)\cdot 19 + \left(10 a^{4} + 11 a^{3} + 10 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(10 a^{4} + 7 a^{3} + 6 a^{2} + 10\right)\cdot 19^{3} + \left(17 a^{4} + 15 a^{3} + 6 a^{2} + 13 a + 18\right)\cdot 19^{4} +O(19^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)(9,10)$ | $-1$ |
$1$ | $5$ | $(1,6,7,3,10)(2,4,8,9,5)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,7,10,6,3)(2,8,5,4,9)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,3,6,10,7)(2,9,4,5,8)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,10,3,7,6)(2,5,9,8,4)$ | $\zeta_{5}^{2}$ |
$1$ | $10$ | $(1,8,6,9,7,5,3,2,10,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$1$ | $10$ | $(1,9,3,4,6,5,10,8,7,2)$ | $-\zeta_{5}^{2}$ |
$1$ | $10$ | $(1,2,7,8,10,5,6,4,3,9)$ | $-\zeta_{5}^{3}$ |
$1$ | $10$ | $(1,4,10,2,3,5,7,9,6,8)$ | $-\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.