Basic invariants
Dimension: | $1$ |
Group: | $C_{14}$ |
Conductor: | \(87\)\(\medspace = 3 \cdot 29 \) |
Artin field: | Galois closure of 14.0.22439994995240462987343.1 |
Galois orbit size: | $6$ |
Smallest permutation container: | $C_{14}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{87}(5,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} + \cdots + 1681 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{7} + 12x + 42 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{6} + 18 a^{5} + 4 a^{4} + 4 a^{3} + 11 a^{2} + 34 a + 25 + \left(22 a^{6} + 45 a^{4} + 13 a^{3} + 28 a^{2} + 12 a + 21\right)\cdot 47 + \left(35 a^{6} + 23 a^{5} + 32 a^{4} + 17 a^{3} + 29 a^{2} + 31 a + 23\right)\cdot 47^{2} + \left(36 a^{6} + 23 a^{5} + 10 a^{4} + 39 a^{3} + 7 a^{2} + 4 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 29 a^{5} + 9 a^{4} + 22 a^{3} + 4 a^{2} + 21 a + 37\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 a^{6} + 44 a^{5} + 17 a^{4} + 41 a^{3} + 39 a^{2} + 27 a + 42 + \left(35 a^{6} + 37 a^{5} + 28 a^{4} + 18 a^{3} + 44 a^{2} + 21 a + 27\right)\cdot 47 + \left(15 a^{6} + 11 a^{5} + 29 a^{4} + 9 a^{3} + 6 a^{2} + 2 a + 8\right)\cdot 47^{2} + \left(23 a^{6} + a^{5} + 2 a^{4} + 23 a^{3} + 43 a^{2} + 15 a + 46\right)\cdot 47^{3} + \left(12 a^{6} + 32 a^{5} + 5 a^{4} + 27 a^{3} + 7 a^{2} + 42 a + 36\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 a^{6} + 12 a^{5} + 41 a^{4} + 28 a^{3} + 45 a^{2} + 45 a + 29 + \left(21 a^{6} + 31 a^{5} + 43 a^{4} + 23 a^{3} + 6 a^{2} + 28 a + 18\right)\cdot 47 + \left(2 a^{6} + 14 a^{5} + 12 a^{4} + 35 a^{3} + 28 a^{2} + 46 a + 46\right)\cdot 47^{2} + \left(25 a^{6} + 18 a^{5} + 14 a^{4} + 20 a^{3} + 46 a^{2} + 34 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 45 a^{5} + 7 a^{4} + 30 a^{3} + 19 a^{2} + 19 a + 21\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{6} + 33 a^{5} + 16 a^{4} + 11 a^{3} + 21 a^{2} + 24 a + 29 + \left(41 a^{6} + 17 a^{5} + 32 a^{4} + 34 a^{3} + 5 a^{2} + 33 a + 29\right)\cdot 47 + \left(16 a^{6} + 35 a^{5} + 13 a^{4} + 18 a^{3} + 44 a^{2} + 38 a + 33\right)\cdot 47^{2} + \left(10 a^{6} + 12 a^{5} + 29 a^{4} + 32 a^{3} + 6 a^{2} + 7 a + 6\right)\cdot 47^{3} + \left(9 a^{6} + 31 a^{4} + 7 a^{3} + 9 a^{2} + 37 a + 30\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 8 a^{6} + 11 a^{5} + 27 a^{4} + 24 a^{3} + 30 a^{2} + 2 a + 14 + \left(21 a^{6} + a^{5} + 20 a^{4} + 8 a^{3} + 21 a^{2} + 38 a + 33\right)\cdot 47 + \left(46 a^{6} + 34 a^{5} + 7 a^{4} + 20 a^{3} + 30 a^{2} + 19 a + 6\right)\cdot 47^{2} + \left(8 a^{6} + 30 a^{5} + 36 a^{4} + 21 a^{3} + 46 a^{2} + 25 a + 31\right)\cdot 47^{3} + \left(9 a^{6} + 4 a^{5} + 42 a^{4} + 45 a^{3} + 36 a + 31\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 22 a^{6} + 41 a^{5} + 28 a^{4} + 6 a^{3} + 44 a^{2} + 25 a + 17 + \left(44 a^{6} + 3 a^{5} + 16 a^{4} + 17 a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 47 + \left(29 a^{6} + 43 a^{5} + 7 a^{4} + 19 a^{3} + 14 a^{2} + 17 a + 45\right)\cdot 47^{2} + \left(43 a^{6} + 39 a^{5} + 35 a^{4} + 41 a^{3} + 30 a^{2} + 9 a + 24\right)\cdot 47^{3} + \left(36 a^{6} + 45 a^{5} + 43 a^{4} + 26 a^{3} + 43 a^{2} + 21 a + 21\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 25 a^{6} + 12 a^{5} + 21 a^{4} + 31 a^{3} + 45 a^{2} + 9 a + 10 + \left(25 a^{6} + 43 a^{5} + 37 a^{4} + 32 a^{3} + 10 a^{2} + 45 a + 44\right)\cdot 47 + \left(44 a^{6} + 9 a^{5} + 41 a^{4} + 9 a^{3} + 8 a^{2} + 45 a + 42\right)\cdot 47^{2} + \left(29 a^{6} + a^{5} + 39 a^{4} + 21 a^{3} + 23 a^{2} + 14 a + 26\right)\cdot 47^{3} + \left(28 a^{6} + 41 a^{5} + 17 a^{4} + a^{3} + 32 a^{2} + 13 a + 1\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 8 }$ | $=$ | \( 25 a^{6} + 20 a^{5} + 33 a^{4} + 16 a^{3} + 22 a^{2} + 15 a + 10 + \left(5 a^{6} + 13 a^{5} + 13 a^{4} + 38 a^{3} + 14 a^{2} + 38 a + 33\right)\cdot 47 + \left(22 a^{6} + 42 a^{5} + 22 a^{4} + 29 a^{3} + 37 a^{2} + 17 a + 13\right)\cdot 47^{2} + \left(35 a^{6} + 31 a^{5} + 4 a^{4} + 15 a^{3} + 37 a^{2} + 32 a + 23\right)\cdot 47^{3} + \left(31 a^{6} + 33 a^{5} + 15 a^{4} + 2 a^{3} + 34 a^{2} + 5 a + 13\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 9 }$ | $=$ | \( 25 a^{6} + 38 a^{5} + 15 a^{4} + 29 a^{3} + 6 a^{2} + 23 a + 21 + \left(37 a^{6} + 11 a^{5} + 7 a^{4} + 40 a^{3} + 14 a^{2} + 2 a + 40\right)\cdot 47 + \left(17 a^{6} + 8 a^{5} + 38 a^{4} + 2 a^{3} + 45 a^{2} + 20\right)\cdot 47^{2} + \left(40 a^{6} + 33 a^{5} + 41 a^{4} + a^{3} + 38 a^{2} + 33 a + 38\right)\cdot 47^{3} + \left(17 a^{6} + 18 a^{5} + 21 a^{4} + 34 a^{3} + 25 a^{2} + 15 a + 26\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 10 }$ | $=$ | \( 26 a^{6} + 3 a^{5} + 39 a^{4} + 30 a^{3} + 46 a^{2} + 39 a + 38 + \left(29 a^{6} + 38 a^{5} + 15 a^{4} + a^{3} + 3 a^{2} + 10 a + 18\right)\cdot 47 + \left(5 a^{6} + 11 a^{5} + 33 a^{4} + 17 a^{3} + 4 a^{2} + 34 a + 43\right)\cdot 47^{2} + \left(26 a^{5} + 31 a^{4} + a^{3} + 39 a^{2} + 44 a + 33\right)\cdot 47^{3} + \left(45 a^{5} + 6 a^{3} + 9 a^{2} + 8 a + 17\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 11 }$ | $=$ | \( 30 a^{6} + a^{5} + 42 a^{4} + 24 a^{3} + a^{2} + 14 a + 12 + \left(35 a^{6} + 16 a^{5} + 5 a^{4} + 27 a^{3} + 30 a^{2} + 5 a + 14\right)\cdot 47 + \left(12 a^{6} + 41 a^{5} + 45 a^{4} + 24 a^{3} + 41 a^{2} + 31 a + 9\right)\cdot 47^{2} + \left(36 a^{6} + 31 a^{5} + 42 a^{4} + 34 a^{3} + 12 a^{2} + 45 a + 43\right)\cdot 47^{3} + \left(32 a^{6} + 18 a^{5} + 40 a^{4} + 7 a^{3} + 41 a^{2} + 21 a + 45\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 12 }$ | $=$ | \( 33 a^{6} + 2 a^{5} + 9 a^{4} + 10 a^{3} + 5 a^{2} + 34 a + 5 + \left(37 a^{6} + 44 a^{5} + 34 a^{4} + 27 a^{3} + 30 a^{2} + 7 a + 35\right)\cdot 47 + \left(3 a^{6} + 3 a^{5} + 34 a^{4} + 20 a^{3} + 33 a^{2} + 5 a + 26\right)\cdot 47^{2} + \left(27 a^{6} + 5 a^{5} + 39 a^{4} + 35 a^{3} + 22 a^{2} + 31 a + 17\right)\cdot 47^{3} + \left(6 a^{6} + 6 a^{5} + 7 a^{4} + a^{3} + 32 a^{2} + a + 16\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 13 }$ | $=$ | \( 37 a^{6} + 28 a^{5} + 10 a^{4} + 34 a^{3} + a^{2} + 38 a + 37 + \left(10 a^{6} + 35 a^{5} + 2 a^{4} + 20 a^{3} + 22 a^{2} + 26 a + 13\right)\cdot 47 + \left(26 a^{6} + 8 a^{5} + 2 a^{4} + 13 a^{3} + 24 a^{2} + 33 a + 20\right)\cdot 47^{2} + \left(20 a^{6} + 21 a^{5} + 3 a^{4} + 4 a^{3} + 23 a^{2} + 28 a + 2\right)\cdot 47^{3} + \left(34 a^{6} + 17 a^{5} + 34 a^{4} + 33 a^{3} + 40 a^{2} + 23 a + 16\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 14 }$ | $=$ | \( 40 a^{6} + 19 a^{5} + 27 a^{4} + 41 a^{3} + 13 a^{2} + 41 + \left(8 a^{6} + 34 a^{5} + 25 a^{4} + 24 a^{3} + 35 a^{2} + 45 a + 40\right)\cdot 47 + \left(2 a^{6} + 40 a^{5} + 7 a^{4} + 43 a^{3} + 27 a^{2} + 4 a + 34\right)\cdot 47^{2} + \left(38 a^{6} + 4 a^{5} + 44 a^{4} + 36 a^{3} + 43 a^{2} + a + 7\right)\cdot 47^{3} + \left(9 a^{6} + 37 a^{5} + 3 a^{4} + 34 a^{3} + 25 a^{2} + 13 a + 11\right)\cdot 47^{4} +O(47^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 14 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 14 }$ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,13)(3,11)(4,14)(6,7)(8,10)(9,12)$ | $-1$ |
$1$ | $7$ | $(1,2,12,3,8,7,4)(5,13,9,11,10,6,14)$ | $\zeta_{7}^{4}$ |
$1$ | $7$ | $(1,12,8,4,2,3,7)(5,9,10,14,13,11,6)$ | $\zeta_{7}$ |
$1$ | $7$ | $(1,3,4,12,7,2,8)(5,11,14,9,6,13,10)$ | $\zeta_{7}^{5}$ |
$1$ | $7$ | $(1,8,2,7,12,4,3)(5,10,13,6,9,14,11)$ | $\zeta_{7}^{2}$ |
$1$ | $7$ | $(1,7,3,2,4,8,12)(5,6,11,13,14,10,9)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$1$ | $7$ | $(1,4,7,8,3,12,2)(5,14,6,10,11,9,13)$ | $\zeta_{7}^{3}$ |
$1$ | $14$ | $(1,13,12,11,8,6,4,5,2,9,3,10,7,14)$ | $-\zeta_{7}^{4}$ |
$1$ | $14$ | $(1,11,4,9,7,13,8,5,3,14,12,6,2,10)$ | $-\zeta_{7}^{5}$ |
$1$ | $14$ | $(1,6,3,13,4,10,12,5,7,11,2,14,8,9)$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ |
$1$ | $14$ | $(1,9,8,14,2,11,7,5,12,10,4,13,3,6)$ | $-\zeta_{7}$ |
$1$ | $14$ | $(1,10,2,6,12,14,3,5,8,13,7,9,4,11)$ | $-\zeta_{7}^{2}$ |
$1$ | $14$ | $(1,14,7,10,3,9,2,5,4,6,8,11,12,13)$ | $-\zeta_{7}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.