Basic invariants
Dimension: | $1$ |
Group: | $C_{12}$ |
Conductor: | \(335\)\(\medspace = 5 \cdot 67 \) |
Artin field: | Galois closure of 12.0.793100932727814453125.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{12}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{335}(163,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + \cdots + 625 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{4} + 5x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 31 a^{3} + 37 a^{2} + 12 a + 21 + \left(7 a^{3} + 10 a^{2} + 1\right)\cdot 43 + \left(29 a^{3} + 22 a^{2} + 37 a + 2\right)\cdot 43^{2} + \left(26 a^{3} + 23 a^{2} + 13 a + 21\right)\cdot 43^{3} + \left(22 a^{3} + 8 a^{2} + 35 a + 16\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 33 a^{3} + 39 a^{2} + 23 a + 23 + \left(5 a^{3} + 28 a^{2} + 10 a + 34\right)\cdot 43 + \left(35 a^{3} + 38 a + 20\right)\cdot 43^{2} + \left(39 a^{3} + 15 a^{2} + 42 a + 2\right)\cdot 43^{3} + \left(17 a^{3} + 31 a^{2} + 29 a + 8\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 7 a^{3} + 20 a^{2} + 14 a + 14 + \left(4 a^{3} + 23 a^{2} + 13 a + 22\right)\cdot 43 + \left(34 a^{3} + 12 a^{2} + 12 a + 37\right)\cdot 43^{2} + \left(38 a^{3} + 9 a^{2} + 4\right)\cdot 43^{3} + \left(33 a^{3} + 35 a^{2} + 37 a + 25\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{3} + 23 a^{2} + 40 a + 27 + \left(37 a^{3} + 27 a^{2} + 3 a + 6\right)\cdot 43 + \left(40 a^{3} + 22 a^{2} + 24 a + 41\right)\cdot 43^{2} + \left(18 a^{3} + 40 a^{2} + 11 a + 10\right)\cdot 43^{3} + \left(18 a^{3} + 5 a^{2} + 33 a + 25\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 5 }$ | $=$ | \( 36 a^{3} + 15 a^{2} + 4 a + 16 + \left(20 a^{3} + 7 a^{2} + 16 a + 40\right)\cdot 43 + \left(17 a^{3} + 11 a^{2} + 3 a + 35\right)\cdot 43^{2} + \left(30 a^{3} + 2 a^{2} + 8 a + 20\right)\cdot 43^{3} + \left(5 a^{3} + 41 a^{2} + 19 a + 16\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 6 }$ | $=$ | \( 28 a^{3} + 14 a^{2} + 15 a + 37 + \left(32 a^{3} + 14 a^{2} + 20 a + 36\right)\cdot 43 + \left(3 a^{3} + 21 a^{2} + 13 a + 6\right)\cdot 43^{2} + \left(7 a^{3} + 19 a^{2} + 6 a + 42\right)\cdot 43^{3} + \left(11 a^{3} + 21 a^{2} + 15 a + 26\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 7 }$ | $=$ | \( 22 a^{3} + 18 a^{2} + 33 a + 34 + \left(41 a^{3} + 4 a^{2} + 33 a + 17\right)\cdot 43 + \left(a^{3} + 37 a^{2} + 23 a + 20\right)\cdot 43^{2} + \left(21 a^{3} + 30 a^{2} + 13 a + 12\right)\cdot 43^{3} + \left(21 a^{3} + 35 a^{2} + 13 a + 27\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 8 }$ | $=$ | \( 34 a^{3} + 7 a^{2} + 42 a + 39 + \left(6 a^{3} + 3 a^{2} + 25 a + 23\right)\cdot 43 + \left(9 a^{3} + 8 a^{2} + 15 a + 16\right)\cdot 43^{2} + \left(40 a^{3} + 32 a^{2} + 38 a + 36\right)\cdot 43^{3} + \left(a^{3} + 40 a^{2} + 35 a\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 9 }$ | $=$ | \( 2 a^{3} + 8 a^{2} + 5 a + 20 + \left(34 a^{2} + 41 a + 5\right)\cdot 43 + \left(7 a^{3} + 17 a^{2} + 19 a + 14\right)\cdot 43^{2} + \left(30 a^{3} + 12 a^{2} + 10 a + 20\right)\cdot 43^{3} + \left(18 a^{3} + 35 a^{2} + 30 a + 19\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 10 }$ | $=$ | \( 16 a^{3} + 17 a^{2} + 24 a + 13 + \left(36 a^{3} + 26 a^{2} + 2 a + 3\right)\cdot 43 + \left(11 a^{2} + 8 a\right)\cdot 43^{2} + \left(30 a^{3} + 41 a^{2} + 36 a + 39\right)\cdot 43^{3} + \left(4 a^{3} + 7 a^{2} + 29 a + 5\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 11 }$ | $=$ | \( 40 a^{3} + 16 a^{2} + 37 a + 37 + \left(15 a^{3} + 20 a^{2} + 35 a + 25\right)\cdot 43 + \left(37 a^{3} + 15 a^{2} + 21 a + 6\right)\cdot 43^{2} + \left(7 a^{3} + 29 a^{2} + 7 a + 2\right)\cdot 43^{3} + \left(36 a^{3} + 18 a + 37\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 12 }$ | $=$ | \( 6 a^{3} + a^{2} + 9 a + 21 + \left(6 a^{3} + 14 a^{2} + 11 a + 39\right)\cdot 43 + \left(41 a^{3} + 34 a^{2} + 40 a + 12\right)\cdot 43^{2} + \left(9 a^{3} + a^{2} + 25 a + 2\right)\cdot 43^{3} + \left(22 a^{3} + 37 a^{2} + 3 a + 6\right)\cdot 43^{4} +O(43^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-1$ |
$1$ | $3$ | $(1,4,6)(2,3,11)(5,8,9)(7,10,12)$ | $-\zeta_{12}^{2}$ |
$1$ | $3$ | $(1,6,4)(2,11,3)(5,9,8)(7,12,10)$ | $\zeta_{12}^{2} - 1$ |
$1$ | $4$ | $(1,5,7,11)(2,4,8,10)(3,6,9,12)$ | $\zeta_{12}^{3}$ |
$1$ | $4$ | $(1,11,7,5)(2,10,8,4)(3,12,9,6)$ | $-\zeta_{12}^{3}$ |
$1$ | $6$ | $(1,12,4,7,6,10)(2,5,3,8,11,9)$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $6$ | $(1,10,6,7,4,12)(2,9,11,8,3,5)$ | $\zeta_{12}^{2}$ |
$1$ | $12$ | $(1,2,12,5,4,3,7,8,6,11,10,9)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$1$ | $12$ | $(1,3,10,5,6,2,7,9,4,11,12,8)$ | $\zeta_{12}$ |
$1$ | $12$ | $(1,8,12,11,4,9,7,2,6,5,10,3)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$1$ | $12$ | $(1,9,10,11,6,8,7,3,4,5,12,2)$ | $-\zeta_{12}$ |
The blue line marks the conjugacy class containing complex conjugation.