Basic invariants
Dimension: | $1$ |
Group: | $C_{14}$ |
Conductor: | \(203\)\(\medspace = 7 \cdot 29 \) |
Artin field: | Galois closure of 14.0.291381688005381590432263.1 |
Galois orbit size: | $6$ |
Smallest permutation container: | $C_{14}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{203}(20,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{14} - 5 x^{13} - x^{12} + 41 x^{11} - 266 x^{9} + 682 x^{8} - 1271 x^{7} + 3346 x^{6} - 5784 x^{5} + \cdots + 22271 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{7} + 21x + 18 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{6} + 13 a^{5} + 21 a^{4} + 17 a^{3} + 15 a^{2} + 19 a + 6 + \left(20 a^{6} + 21 a^{5} + 12 a^{4} + 16 a^{2} + 6 a + 6\right)\cdot 23 + \left(2 a^{6} + 8 a^{5} + 2 a^{4} + 6 a^{3} + 14 a^{2} + 16 a + 21\right)\cdot 23^{2} + \left(18 a^{6} + 13 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 4 a + 14\right)\cdot 23^{3} + \left(10 a^{6} + 2 a^{5} + 17 a^{4} + 9 a^{3} + 6 a^{2} + 22 a + 10\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 a^{6} + 13 a^{5} + 21 a^{4} + 17 a^{3} + 15 a^{2} + 19 a + 10 + \left(20 a^{6} + 21 a^{5} + 12 a^{4} + 16 a^{2} + 6 a + 3\right)\cdot 23 + \left(2 a^{6} + 8 a^{5} + 2 a^{4} + 6 a^{3} + 14 a^{2} + 16 a + 20\right)\cdot 23^{2} + \left(18 a^{6} + 13 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 4 a + 19\right)\cdot 23^{3} + \left(10 a^{6} + 2 a^{5} + 17 a^{4} + 9 a^{3} + 6 a^{2} + 22 a + 22\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{6} + 13 a^{5} + 7 a^{4} + 15 a^{3} + 19 a^{2} + a + 1 + \left(22 a^{6} + 21 a^{5} + 10 a^{4} + 3 a^{3} + 6 a^{2} + 16 a + 20\right)\cdot 23 + \left(18 a^{6} + 4 a^{5} + 4 a^{4} + 14 a^{3} + 13 a + 11\right)\cdot 23^{2} + \left(17 a^{6} + 22 a^{5} + 8 a^{4} + 22 a^{3} + 9 a^{2} + 4 a + 9\right)\cdot 23^{3} + \left(5 a^{6} + 21 a^{5} + 22 a^{4} + 2 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( 9 a^{6} + 13 a^{5} + 7 a^{4} + 15 a^{3} + 19 a^{2} + a + 5 + \left(22 a^{6} + 21 a^{5} + 10 a^{4} + 3 a^{3} + 6 a^{2} + 16 a + 17\right)\cdot 23 + \left(18 a^{6} + 4 a^{5} + 4 a^{4} + 14 a^{3} + 13 a + 10\right)\cdot 23^{2} + \left(17 a^{6} + 22 a^{5} + 8 a^{4} + 22 a^{3} + 9 a^{2} + 4 a + 14\right)\cdot 23^{3} + \left(5 a^{6} + 21 a^{5} + 22 a^{4} + 2 a^{3} + 7 a^{2} + 9 a + 1\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 10 a^{6} + 9 a^{5} + 19 a^{4} + 2 a^{3} + 10 a^{2} + 10 a + \left(11 a^{6} + 16 a^{5} + 16 a^{4} + 17 a^{3} + a^{2} + 15 a + 4\right)\cdot 23 + \left(21 a^{6} + 19 a^{5} + 2 a^{4} + 7 a^{3} + 9 a^{2} + 10\right)\cdot 23^{2} + \left(2 a^{6} + 2 a^{5} + 12 a^{4} + 16 a^{3} + 12 a^{2} + 22\right)\cdot 23^{3} + \left(7 a^{6} + 18 a^{5} + a^{4} + 3 a^{3} + 12 a^{2} + 9 a + 2\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 6 }$ | $=$ | \( 10 a^{6} + 9 a^{5} + 19 a^{4} + 2 a^{3} + 10 a^{2} + 10 a + 19 + \left(11 a^{6} + 16 a^{5} + 16 a^{4} + 17 a^{3} + a^{2} + 15 a + 6\right)\cdot 23 + \left(21 a^{6} + 19 a^{5} + 2 a^{4} + 7 a^{3} + 9 a^{2} + 11\right)\cdot 23^{2} + \left(2 a^{6} + 2 a^{5} + 12 a^{4} + 16 a^{3} + 12 a^{2} + 17\right)\cdot 23^{3} + \left(7 a^{6} + 18 a^{5} + a^{4} + 3 a^{3} + 12 a^{2} + 9 a + 13\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{6} + 6 a^{5} + 6 a^{4} + a^{3} + 20 a^{2} + 16 a + 17 + \left(3 a^{6} + 20 a^{5} + 8 a^{4} + 21 a^{3} + 10 a^{2} + 16 a + 4\right)\cdot 23 + \left(7 a^{6} + 9 a^{5} + 18 a^{4} + 3 a^{3} + 17 a^{2} + 22 a + 6\right)\cdot 23^{2} + \left(8 a^{6} + 12 a^{5} + 18 a^{4} + 19 a^{3} + 19 a^{2} + 8 a + 22\right)\cdot 23^{3} + \left(20 a^{6} + 6 a^{5} + 14 a^{4} + a^{3} + 13 a^{2} + 17 a + 21\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 8 }$ | $=$ | \( 15 a^{6} + 6 a^{5} + 6 a^{4} + a^{3} + 20 a^{2} + 16 a + 21 + \left(3 a^{6} + 20 a^{5} + 8 a^{4} + 21 a^{3} + 10 a^{2} + 16 a + 1\right)\cdot 23 + \left(7 a^{6} + 9 a^{5} + 18 a^{4} + 3 a^{3} + 17 a^{2} + 22 a + 5\right)\cdot 23^{2} + \left(8 a^{6} + 12 a^{5} + 18 a^{4} + 19 a^{3} + 19 a^{2} + 8 a + 4\right)\cdot 23^{3} + \left(20 a^{6} + 6 a^{5} + 14 a^{4} + a^{3} + 13 a^{2} + 17 a + 11\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 9 }$ | $=$ | \( 15 a^{6} + 12 a^{5} + 16 a^{4} + 2 a^{3} + 22 a^{2} + 9 a + 17 + \left(22 a^{6} + 19 a^{5} + 8 a^{4} + 19 a^{3} + 18 a^{2} + 21 a + 1\right)\cdot 23 + \left(13 a^{6} + 9 a^{5} + 4 a^{4} + 5 a^{3} + 12 a^{2} + 12 a + 14\right)\cdot 23^{2} + \left(17 a^{6} + 11 a^{4} + 19 a^{3} + 16 a^{2} + 20 a + 5\right)\cdot 23^{3} + \left(19 a^{6} + 14 a^{5} + 15 a^{3} + 9 a^{2} + 18 a + 11\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 10 }$ | $=$ | \( 15 a^{6} + 12 a^{5} + 16 a^{4} + 2 a^{3} + 22 a^{2} + 9 a + 21 + \left(22 a^{6} + 19 a^{5} + 8 a^{4} + 19 a^{3} + 18 a^{2} + 21 a + 21\right)\cdot 23 + \left(13 a^{6} + 9 a^{5} + 4 a^{4} + 5 a^{3} + 12 a^{2} + 12 a + 12\right)\cdot 23^{2} + \left(17 a^{6} + 11 a^{4} + 19 a^{3} + 16 a^{2} + 20 a + 10\right)\cdot 23^{3} + \left(19 a^{6} + 14 a^{5} + 15 a^{3} + 9 a^{2} + 18 a\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 11 }$ | $=$ | \( 17 a^{6} + 15 a^{5} + 18 a^{4} + 12 a^{3} + 5 a^{2} + 11 a + 7 + \left(16 a^{6} + 18 a^{5} + 13 a^{4} + 8 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 23 + \left(22 a^{6} + 14 a^{5} + 16 a^{4} + 14 a^{3} + 11 a^{2} + 16 a + 10\right)\cdot 23^{2} + \left(22 a^{6} + 5 a^{5} + 10 a^{3} + 4 a^{2} + 22 a + 10\right)\cdot 23^{3} + \left(10 a^{6} + 6 a^{5} + 19 a^{4} + 22 a^{3} + 3 a^{2} + 3 a + 14\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 12 }$ | $=$ | \( 17 a^{6} + 15 a^{5} + 18 a^{4} + 12 a^{3} + 5 a^{2} + 11 a + 11 + \left(16 a^{6} + 18 a^{5} + 13 a^{4} + 8 a^{3} + 12 a^{2} + 6 a + 7\right)\cdot 23 + \left(22 a^{6} + 14 a^{5} + 16 a^{4} + 14 a^{3} + 11 a^{2} + 16 a + 9\right)\cdot 23^{2} + \left(22 a^{6} + 5 a^{5} + 10 a^{3} + 4 a^{2} + 22 a + 15\right)\cdot 23^{3} + \left(10 a^{6} + 6 a^{5} + 19 a^{4} + 22 a^{3} + 3 a^{2} + 3 a + 3\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 13 }$ | $=$ | \( 18 a^{6} + a^{5} + 5 a^{4} + 20 a^{3} + a^{2} + 3 a + 2 + \left(17 a^{6} + 20 a^{5} + 21 a^{4} + 21 a^{3} + 2 a^{2} + 9 a + 6\right)\cdot 23 + \left(4 a^{6} + 19 a^{4} + 16 a^{3} + 3 a^{2} + 9 a + 9\right)\cdot 23^{2} + \left(4 a^{6} + 12 a^{5} + 6 a^{4} + 4 a^{3} + 16 a^{2} + 7 a + 17\right)\cdot 23^{3} + \left(17 a^{6} + 22 a^{5} + 16 a^{4} + 12 a^{3} + 15 a^{2} + 11 a + 10\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 14 }$ | $=$ | \( 18 a^{6} + a^{5} + 5 a^{4} + 20 a^{3} + a^{2} + 3 a + 6 + \left(17 a^{6} + 20 a^{5} + 21 a^{4} + 21 a^{3} + 2 a^{2} + 9 a + 3\right)\cdot 23 + \left(4 a^{6} + 19 a^{4} + 16 a^{3} + 3 a^{2} + 9 a + 8\right)\cdot 23^{2} + \left(4 a^{6} + 12 a^{5} + 6 a^{4} + 4 a^{3} + 16 a^{2} + 7 a + 22\right)\cdot 23^{3} + \left(17 a^{6} + 22 a^{5} + 16 a^{4} + 12 a^{3} + 15 a^{2} + 11 a + 22\right)\cdot 23^{4} +O(23^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)$ | $-1$ |
$1$ | $7$ | $(1,6,3,7,11,13,9)(2,5,4,8,12,14,10)$ | $\zeta_{7}^{2}$ |
$1$ | $7$ | $(1,3,11,9,6,7,13)(2,4,12,10,5,8,14)$ | $\zeta_{7}^{4}$ |
$1$ | $7$ | $(1,7,9,3,13,6,11)(2,8,10,4,14,5,12)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$1$ | $7$ | $(1,11,6,13,3,9,7)(2,12,5,14,4,10,8)$ | $\zeta_{7}$ |
$1$ | $7$ | $(1,13,7,6,9,11,3)(2,14,8,5,10,12,4)$ | $\zeta_{7}^{3}$ |
$1$ | $7$ | $(1,9,13,11,7,3,6)(2,10,14,12,8,4,5)$ | $\zeta_{7}^{5}$ |
$1$ | $14$ | $(1,5,3,8,11,14,9,2,6,4,7,12,13,10)$ | $-\zeta_{7}^{2}$ |
$1$ | $14$ | $(1,8,9,4,13,5,11,2,7,10,3,14,6,12)$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ |
$1$ | $14$ | $(1,14,7,5,9,12,3,2,13,8,6,10,11,4)$ | $-\zeta_{7}^{3}$ |
$1$ | $14$ | $(1,4,11,10,6,8,13,2,3,12,9,5,7,14)$ | $-\zeta_{7}^{4}$ |
$1$ | $14$ | $(1,12,6,14,3,10,7,2,11,5,13,4,9,8)$ | $-\zeta_{7}$ |
$1$ | $14$ | $(1,10,13,12,7,4,6,2,9,14,11,8,3,5)$ | $-\zeta_{7}^{5}$ |
The blue line marks the conjugacy class containing complex conjugation.