Basic invariants
Dimension: | $1$ |
Group: | $C_{12}$ |
Conductor: | \(285\)\(\medspace = 3 \cdot 5 \cdot 19 \) |
Artin field: | Galois closure of 12.12.24181674720486328125.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{12}$ |
Parity: | even |
Dirichlet character: | \(\chi_{285}(182,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} - 38 x^{10} + 46 x^{9} + 447 x^{8} - 632 x^{7} - 1642 x^{6} + 2478 x^{5} + 1037 x^{4} + \cdots + 31 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{4} + 6x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a^{3} + 36 a^{2} + 30 a + 28 + \left(12 a^{3} + 24 a^{2} + 18 a + 35\right)\cdot 37 + \left(16 a^{3} + 30 a^{2} + 13 a + 19\right)\cdot 37^{2} + \left(10 a^{3} + 30 a^{2} + 6 a + 30\right)\cdot 37^{3} + \left(33 a^{3} + 11 a^{2} + 16 a + 30\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 a^{3} + 36 a^{2} + 30 a + 12 + \left(12 a^{3} + 24 a^{2} + 18 a + 30\right)\cdot 37 + \left(16 a^{3} + 30 a^{2} + 13 a + 20\right)\cdot 37^{2} + \left(10 a^{3} + 30 a^{2} + 6 a + 21\right)\cdot 37^{3} + \left(33 a^{3} + 11 a^{2} + 16 a\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{3} + 36 a^{2} + 30 a + 13 + \left(12 a^{3} + 24 a^{2} + 18 a + 23\right)\cdot 37 + \left(16 a^{3} + 30 a^{2} + 13 a + 34\right)\cdot 37^{2} + \left(10 a^{3} + 30 a^{2} + 6 a + 2\right)\cdot 37^{3} + \left(33 a^{3} + 11 a^{2} + 16 a + 13\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( a^{3} + 11 a^{2} + 22 a + 9 + \left(7 a^{3} + 20 a^{2} + 3 a + 5\right)\cdot 37 + \left(35 a^{3} + 36 a^{2} + 24 a + 7\right)\cdot 37^{2} + \left(21 a^{3} + 26 a^{2} + 12 a + 4\right)\cdot 37^{3} + \left(3 a^{3} + 18 a^{2} + 36 a + 35\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( a^{3} + 11 a^{2} + 22 a + 30 + \left(7 a^{3} + 20 a^{2} + 3 a + 36\right)\cdot 37 + \left(35 a^{3} + 36 a^{2} + 24 a + 7\right)\cdot 37^{2} + \left(21 a^{3} + 26 a^{2} + 12 a + 32\right)\cdot 37^{3} + \left(3 a^{3} + 18 a^{2} + 36 a + 4\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 6 }$ | $=$ | \( a^{3} + 11 a^{2} + 22 a + 31 + \left(7 a^{3} + 20 a^{2} + 3 a + 29\right)\cdot 37 + \left(35 a^{3} + 36 a^{2} + 24 a + 21\right)\cdot 37^{2} + \left(21 a^{3} + 26 a^{2} + 12 a + 13\right)\cdot 37^{3} + \left(3 a^{3} + 18 a^{2} + 36 a + 17\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 7 }$ | $=$ | \( 33 a^{3} + 13 a^{2} + 36 a + 20 + \left(14 a^{3} + 23 a^{2} + 12 a + 2\right)\cdot 37 + \left(23 a^{3} + 33 a^{2} + 27 a + 9\right)\cdot 37^{2} + \left(27 a^{3} + 21 a^{2} + 5 a + 8\right)\cdot 37^{3} + \left(7 a^{3} + 13 a^{2} + 34 a + 27\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 8 }$ | $=$ | \( 33 a^{3} + 13 a^{2} + 36 a + 21 + \left(14 a^{3} + 23 a^{2} + 12 a + 32\right)\cdot 37 + \left(23 a^{3} + 33 a^{2} + 27 a + 22\right)\cdot 37^{2} + \left(27 a^{3} + 21 a^{2} + 5 a + 26\right)\cdot 37^{3} + \left(7 a^{3} + 13 a^{2} + 34 a + 2\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 9 }$ | $=$ | \( 33 a^{3} + 13 a^{2} + 36 a + 36 + \left(14 a^{3} + 23 a^{2} + 12 a + 7\right)\cdot 37 + \left(23 a^{3} + 33 a^{2} + 27 a + 8\right)\cdot 37^{2} + \left(27 a^{3} + 21 a^{2} + 5 a + 17\right)\cdot 37^{3} + \left(7 a^{3} + 13 a^{2} + 34 a + 20\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 10 }$ | $=$ | \( a^{3} + 14 a^{2} + 23 a + 18 + \left(3 a^{3} + 5 a^{2} + a + 36\right)\cdot 37 + \left(36 a^{3} + 10 a^{2} + 9 a + 17\right)\cdot 37^{2} + \left(13 a^{3} + 31 a^{2} + 12 a + 21\right)\cdot 37^{3} + \left(29 a^{3} + 29 a^{2} + 24 a + 14\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 11 }$ | $=$ | \( a^{3} + 14 a^{2} + 23 a + 3 + \left(3 a^{3} + 5 a^{2} + a + 24\right)\cdot 37 + \left(36 a^{3} + 10 a^{2} + 9 a + 32\right)\cdot 37^{2} + \left(13 a^{3} + 31 a^{2} + 12 a + 30\right)\cdot 37^{3} + \left(29 a^{3} + 29 a^{2} + 24 a + 33\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 12 }$ | $=$ | \( a^{3} + 14 a^{2} + 23 a + 2 + \left(3 a^{3} + 5 a^{2} + a + 31\right)\cdot 37 + \left(36 a^{3} + 10 a^{2} + 9 a + 18\right)\cdot 37^{2} + \left(13 a^{3} + 31 a^{2} + 12 a + 12\right)\cdot 37^{3} + \left(29 a^{3} + 29 a^{2} + 24 a + 21\right)\cdot 37^{4} +O(37^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,9)(2,7)(3,8)(4,10)(5,12)(6,11)$ | $-1$ |
$1$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)(10,12,11)$ | $-\zeta_{12}^{2}$ |
$1$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)(10,11,12)$ | $\zeta_{12}^{2} - 1$ |
$1$ | $4$ | $(1,10,9,4)(2,12,7,5)(3,11,8,6)$ | $\zeta_{12}^{3}$ |
$1$ | $4$ | $(1,4,9,10)(2,5,7,12)(3,6,8,11)$ | $-\zeta_{12}^{3}$ |
$1$ | $6$ | $(1,8,2,9,3,7)(4,11,5,10,6,12)$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $6$ | $(1,7,3,9,2,8)(4,12,6,10,5,11)$ | $\zeta_{12}^{2}$ |
$1$ | $12$ | $(1,5,8,10,2,6,9,12,3,4,7,11)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$1$ | $12$ | $(1,6,7,10,3,5,9,11,2,4,8,12)$ | $\zeta_{12}$ |
$1$ | $12$ | $(1,12,8,4,2,11,9,5,3,10,7,6)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$1$ | $12$ | $(1,11,7,4,3,12,9,6,2,10,8,5)$ | $-\zeta_{12}$ |
The blue line marks the conjugacy class containing complex conjugation.