Basic invariants
Dimension: | $1$ |
Group: | $C_{12}$ |
Conductor: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
Artin field: | Galois closure of 12.0.3099363912000000000.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{12}$ |
Parity: | odd |
Dirichlet character: | \(\chi_{180}(83,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} + 30x^{10} + 315x^{8} + 1500x^{6} + 3375x^{4} + 3375x^{2} + 1125 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{4} + 6x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 24 a^{2} + 8 a + 35 + \left(5 a^{3} + 24 a + 17\right)\cdot 37 + \left(32 a^{3} + 3 a^{2} + 21 a + 32\right)\cdot 37^{2} + \left(12 a^{3} + 3 a + 9\right)\cdot 37^{3} + \left(11 a^{3} + 10 a^{2} + 28 a + 12\right)\cdot 37^{4} + \left(21 a^{3} + 36 a + 14\right)\cdot 37^{5} + \left(23 a^{3} + 22 a^{2} + 22 a + 9\right)\cdot 37^{6} + \left(14 a^{3} + 28 a^{2} + 18 a + 16\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 2 }$ | $=$ | \( 20 a^{3} + 15 a^{2} + 17 a + 35 + \left(24 a^{3} + 12 a^{2} + 22 a + 34\right)\cdot 37 + \left(24 a^{3} + a^{2} + 33 a + 3\right)\cdot 37^{2} + \left(13 a^{3} + 27 a^{2} + 35 a + 31\right)\cdot 37^{3} + \left(21 a^{3} + 31 a^{2} + 36 a + 35\right)\cdot 37^{4} + \left(6 a^{3} + 15 a^{2} + 13 a + 17\right)\cdot 37^{5} + \left(35 a^{3} + 3 a^{2} + 7 a + 14\right)\cdot 37^{6} + \left(21 a^{3} + 5 a^{2} + 34 a + 3\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 3 }$ | $=$ | \( 6 a^{3} + 23 a^{2} + 31 a + 29 + \left(17 a^{3} + 16 a^{2} + 22 a + 25\right)\cdot 37 + \left(3 a^{3} + 17 a^{2} + 22 a + 3\right)\cdot 37^{2} + \left(18 a^{3} + 9 a^{2} + 16 a + 21\right)\cdot 37^{3} + \left(11 a^{3} + 29 a^{2} + 22 a + 35\right)\cdot 37^{4} + \left(21 a^{3} + 21 a^{2} + 6 a + 4\right)\cdot 37^{5} + \left(2 a^{3} + 5 a^{2} + 15 a + 26\right)\cdot 37^{6} + \left(28 a^{3} + 33 a^{2} + 21 a + 12\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{3} + 36 a^{2} + 26 a + 10 + \left(32 a^{3} + 7 a^{2} + 28 a + 13\right)\cdot 37 + \left(8 a^{3} + 18 a^{2} + 17 a + 29\right)\cdot 37^{2} + \left(5 a^{3} + 21 a + 21\right)\cdot 37^{3} + \left(4 a^{3} + 13 a^{2} + 14 a + 2\right)\cdot 37^{4} + \left(9 a^{3} + 36 a^{2} + 16 a + 14\right)\cdot 37^{5} + \left(36 a^{3} + 27 a^{2} + 14 a + 33\right)\cdot 37^{6} + \left(23 a^{3} + 35 a^{2} + 18 a + 20\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 5 }$ | $=$ | \( 27 a^{2} + 9 a + 7 + \left(a^{3} + 32 a^{2} + 13 a + 5\right)\cdot 37 + \left(32 a^{3} + 10 a^{2} + 3 a + 17\right)\cdot 37^{2} + \left(6 a^{3} + 30 a^{2} + 5 a + 29\right)\cdot 37^{3} + \left(13 a^{3} + 2 a^{2} + 15 a + 23\right)\cdot 37^{4} + \left(31 a^{3} + 35 a^{2} + 22 a + 3\right)\cdot 37^{5} + \left(19 a^{3} + 14 a^{2} + 2 a + 32\right)\cdot 37^{6} + \left(18 a^{3} + 21 a^{2} + a + 27\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 6 }$ | $=$ | \( 34 a^{2} + 36 a + 28 + \left(4 a^{3} + 4 a^{2} + 10 a + 12\right)\cdot 37 + \left(29 a^{2} + 18 a + 15\right)\cdot 37^{2} + \left(6 a^{3} + 6 a^{2} + 35 a + 17\right)\cdot 37^{3} + \left(35 a^{3} + 7 a^{2} + 12 a + 25\right)\cdot 37^{4} + \left(26 a^{3} + 2 a^{2} + 14 a + 10\right)\cdot 37^{5} + \left(3 a^{3} + 7 a^{2} + 20 a + 14\right)\cdot 37^{6} + \left(33 a^{3} + 7 a^{2} + 17 a + 25\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 7 }$ | $=$ | \( 13 a^{2} + 29 a + 2 + \left(32 a^{3} + 36 a^{2} + 12 a + 19\right)\cdot 37 + \left(4 a^{3} + 33 a^{2} + 15 a + 4\right)\cdot 37^{2} + \left(24 a^{3} + 36 a^{2} + 33 a + 27\right)\cdot 37^{3} + \left(25 a^{3} + 26 a^{2} + 8 a + 24\right)\cdot 37^{4} + \left(15 a^{3} + 36 a^{2} + 22\right)\cdot 37^{5} + \left(13 a^{3} + 14 a^{2} + 14 a + 27\right)\cdot 37^{6} + \left(22 a^{3} + 8 a^{2} + 18 a + 20\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 8 }$ | $=$ | \( 17 a^{3} + 22 a^{2} + 20 a + 2 + \left(12 a^{3} + 24 a^{2} + 14 a + 2\right)\cdot 37 + \left(12 a^{3} + 35 a^{2} + 3 a + 33\right)\cdot 37^{2} + \left(23 a^{3} + 9 a^{2} + a + 5\right)\cdot 37^{3} + \left(15 a^{3} + 5 a^{2} + 1\right)\cdot 37^{4} + \left(30 a^{3} + 21 a^{2} + 23 a + 19\right)\cdot 37^{5} + \left(a^{3} + 33 a^{2} + 29 a + 22\right)\cdot 37^{6} + \left(15 a^{3} + 31 a^{2} + 2 a + 33\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 9 }$ | $=$ | \( 31 a^{3} + 14 a^{2} + 6 a + 8 + \left(19 a^{3} + 20 a^{2} + 14 a + 11\right)\cdot 37 + \left(33 a^{3} + 19 a^{2} + 14 a + 33\right)\cdot 37^{2} + \left(18 a^{3} + 27 a^{2} + 20 a + 15\right)\cdot 37^{3} + \left(25 a^{3} + 7 a^{2} + 14 a + 1\right)\cdot 37^{4} + \left(15 a^{3} + 15 a^{2} + 30 a + 32\right)\cdot 37^{5} + \left(34 a^{3} + 31 a^{2} + 21 a + 10\right)\cdot 37^{6} + \left(8 a^{3} + 3 a^{2} + 15 a + 24\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 10 }$ | $=$ | \( 26 a^{3} + a^{2} + 11 a + 27 + \left(4 a^{3} + 29 a^{2} + 8 a + 23\right)\cdot 37 + \left(28 a^{3} + 18 a^{2} + 19 a + 7\right)\cdot 37^{2} + \left(31 a^{3} + 36 a^{2} + 15 a + 15\right)\cdot 37^{3} + \left(32 a^{3} + 23 a^{2} + 22 a + 34\right)\cdot 37^{4} + \left(27 a^{3} + 20 a + 22\right)\cdot 37^{5} + \left(9 a^{2} + 22 a + 3\right)\cdot 37^{6} + \left(13 a^{3} + a^{2} + 18 a + 16\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 11 }$ | $=$ | \( 10 a^{2} + 28 a + 30 + \left(36 a^{3} + 4 a^{2} + 23 a + 31\right)\cdot 37 + \left(4 a^{3} + 26 a^{2} + 33 a + 19\right)\cdot 37^{2} + \left(30 a^{3} + 6 a^{2} + 31 a + 7\right)\cdot 37^{3} + \left(23 a^{3} + 34 a^{2} + 21 a + 13\right)\cdot 37^{4} + \left(5 a^{3} + a^{2} + 14 a + 33\right)\cdot 37^{5} + \left(17 a^{3} + 22 a^{2} + 34 a + 4\right)\cdot 37^{6} + \left(18 a^{3} + 15 a^{2} + 35 a + 9\right)\cdot 37^{7} +O(37^{8})\) |
$r_{ 12 }$ | $=$ | \( 3 a^{2} + a + 9 + \left(33 a^{3} + 32 a^{2} + 26 a + 24\right)\cdot 37 + \left(36 a^{3} + 7 a^{2} + 18 a + 21\right)\cdot 37^{2} + \left(30 a^{3} + 30 a^{2} + a + 19\right)\cdot 37^{3} + \left(a^{3} + 29 a^{2} + 24 a + 11\right)\cdot 37^{4} + \left(10 a^{3} + 34 a^{2} + 22 a + 26\right)\cdot 37^{5} + \left(33 a^{3} + 29 a^{2} + 16 a + 22\right)\cdot 37^{6} + \left(3 a^{3} + 29 a^{2} + 19 a + 11\right)\cdot 37^{7} +O(37^{8})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-1$ |
$1$ | $3$ | $(1,12,11)(2,4,3)(5,7,6)(8,10,9)$ | $-\zeta_{12}^{2}$ |
$1$ | $3$ | $(1,11,12)(2,3,4)(5,6,7)(8,9,10)$ | $\zeta_{12}^{2} - 1$ |
$1$ | $4$ | $(1,10,7,4)(2,11,8,5)(3,12,9,6)$ | $-\zeta_{12}^{3}$ |
$1$ | $4$ | $(1,4,7,10)(2,5,8,11)(3,6,9,12)$ | $\zeta_{12}^{3}$ |
$1$ | $6$ | $(1,5,12,7,11,6)(2,9,4,8,3,10)$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $6$ | $(1,6,11,7,12,5)(2,10,3,8,4,9)$ | $\zeta_{12}^{2}$ |
$1$ | $12$ | $(1,3,5,10,12,2,7,9,11,4,6,8)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$1$ | $12$ | $(1,2,6,10,11,3,7,8,12,4,5,9)$ | $-\zeta_{12}$ |
$1$ | $12$ | $(1,9,5,4,12,8,7,3,11,10,6,2)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$1$ | $12$ | $(1,8,6,4,11,9,7,2,12,10,5,3)$ | $\zeta_{12}$ |
The blue line marks the conjugacy class containing complex conjugation.