Basic invariants
Dimension: | $1$ |
Group: | $C_{12}$ |
Conductor: | \(465\)\(\medspace = 3 \cdot 5 \cdot 31 \) |
Artin field: | Galois closure of 12.12.1214370246668923828125.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_{12}$ |
Parity: | even |
Dirichlet character: | \(\chi_{465}(377,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} - 54 x^{10} + 58 x^{9} + 948 x^{8} - 1163 x^{7} - 6275 x^{6} + 7932 x^{5} + 14751 x^{4} + \cdots + 5851 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 20 a^{3} + 20 a^{2} + 18 a + 22 + \left(4 a^{3} + 11 a^{2} + 11 a + 16\right)\cdot 23 + \left(10 a^{3} + 14 a^{2} + 2 a + 5\right)\cdot 23^{2} + \left(13 a^{3} + 5 a^{2} + 22 a + 14\right)\cdot 23^{3} + \left(8 a^{2} + 22 a + 18\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 20 a^{3} + 20 a^{2} + 18 a + 4 + \left(4 a^{3} + 11 a^{2} + 11 a + 15\right)\cdot 23 + \left(10 a^{3} + 14 a^{2} + 2 a + 10\right)\cdot 23^{2} + \left(13 a^{3} + 5 a^{2} + 22 a + 4\right)\cdot 23^{3} + \left(8 a^{2} + 22 a + 2\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 13 a^{3} + 2 a + 19 + \left(6 a^{3} + 14 a^{2} + 9 a + 3\right)\cdot 23 + \left(19 a^{3} + 19 a^{2} + 10 a + 7\right)\cdot 23^{2} + \left(17 a^{3} + 15 a^{2} + 21 a + 15\right)\cdot 23^{3} + \left(14 a^{2} + 6 a + 7\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{3} + 21 a^{2} + 4 a + 2 + \left(12 a^{3} + 9 a^{2} + 11 a\right)\cdot 23 + \left(18 a^{3} + 10 a^{2} + 12 a + 8\right)\cdot 23^{2} + \left(22 a^{3} + 11 a^{2} + 14 a + 14\right)\cdot 23^{3} + \left(19 a^{3} + 17 a^{2} + 14 a + 16\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 11 a^{3} + 5 a^{2} + 22 a + 21 + \left(22 a^{3} + 10 a^{2} + 13 a + 17\right)\cdot 23 + \left(20 a^{3} + a^{2} + 20 a + 9\right)\cdot 23^{2} + \left(14 a^{3} + 13 a^{2} + 10 a + 15\right)\cdot 23^{3} + \left(a^{3} + 5 a^{2} + a\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 6 }$ | $=$ | \( 11 a^{3} + 5 a^{2} + 22 a + 15 + \left(22 a^{3} + 10 a^{2} + 13 a + 6\right)\cdot 23 + \left(20 a^{3} + a^{2} + 20 a + 7\right)\cdot 23^{2} + \left(14 a^{3} + 13 a^{2} + 10 a\right)\cdot 23^{3} + \left(a^{3} + 5 a^{2} + a + 1\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 7 }$ | $=$ | \( 13 a^{3} + 2 a + 18 + \left(6 a^{3} + 14 a^{2} + 9 a + 13\right)\cdot 23 + \left(19 a^{3} + 19 a^{2} + 10 a + 9\right)\cdot 23^{2} + \left(17 a^{3} + 15 a^{2} + 21 a + 13\right)\cdot 23^{3} + \left(14 a^{2} + 6 a + 14\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 8 }$ | $=$ | \( 11 a^{3} + 5 a^{2} + 22 a + 20 + \left(22 a^{3} + 10 a^{2} + 13 a + 4\right)\cdot 23 + \left(20 a^{3} + a^{2} + 20 a + 12\right)\cdot 23^{2} + \left(14 a^{3} + 13 a^{2} + 10 a + 13\right)\cdot 23^{3} + \left(a^{3} + 5 a^{2} + a + 7\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 9 }$ | $=$ | \( 2 a^{3} + 21 a^{2} + 4 a + 20 + \left(12 a^{3} + 9 a^{2} + 11 a + 1\right)\cdot 23 + \left(18 a^{3} + 10 a^{2} + 12 a + 3\right)\cdot 23^{2} + \left(22 a^{3} + 11 a^{2} + 14 a + 1\right)\cdot 23^{3} + \left(19 a^{3} + 17 a^{2} + 14 a + 10\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 10 }$ | $=$ | \( 20 a^{3} + 20 a^{2} + 18 a + 5 + \left(4 a^{3} + 11 a^{2} + 11 a + 5\right)\cdot 23 + \left(10 a^{3} + 14 a^{2} + 2 a + 8\right)\cdot 23^{2} + \left(13 a^{3} + 5 a^{2} + 22 a + 6\right)\cdot 23^{3} + \left(8 a^{2} + 22 a + 18\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 11 }$ | $=$ | \( 13 a^{3} + 2 a + 13 + \left(6 a^{3} + 14 a^{2} + 9 a + 15\right)\cdot 23 + \left(19 a^{3} + 19 a^{2} + 10 a + 4\right)\cdot 23^{2} + \left(17 a^{3} + 15 a^{2} + 21 a\right)\cdot 23^{3} + \left(14 a^{2} + 6 a + 8\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 12 }$ | $=$ | \( 2 a^{3} + 21 a^{2} + 4 a + 3 + \left(12 a^{3} + 9 a^{2} + 11 a + 13\right)\cdot 23 + \left(18 a^{3} + 10 a^{2} + 12 a + 5\right)\cdot 23^{2} + \left(22 a^{3} + 11 a^{2} + 14 a + 16\right)\cdot 23^{3} + \left(19 a^{3} + 17 a^{2} + 14 a + 9\right)\cdot 23^{4} +O(23^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,11)(2,7)(3,10)(4,8)(5,12)(6,9)$ | $-1$ |
$1$ | $3$ | $(1,2,10)(3,11,7)(4,12,9)(5,6,8)$ | $\zeta_{12}^{2} - 1$ |
$1$ | $3$ | $(1,10,2)(3,7,11)(4,9,12)(5,8,6)$ | $-\zeta_{12}^{2}$ |
$1$ | $4$ | $(1,9,11,6)(2,4,7,8)(3,5,10,12)$ | $-\zeta_{12}^{3}$ |
$1$ | $4$ | $(1,6,11,9)(2,8,7,4)(3,12,10,5)$ | $\zeta_{12}^{3}$ |
$1$ | $6$ | $(1,3,2,11,10,7)(4,6,12,8,9,5)$ | $\zeta_{12}^{2}$ |
$1$ | $6$ | $(1,7,10,11,2,3)(4,5,9,8,12,6)$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $12$ | $(1,8,3,9,2,5,11,4,10,6,7,12)$ | $-\zeta_{12}$ |
$1$ | $12$ | $(1,5,7,9,10,8,11,12,2,6,3,4)$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$1$ | $12$ | $(1,4,3,6,2,12,11,8,10,9,7,5)$ | $\zeta_{12}$ |
$1$ | $12$ | $(1,12,7,6,10,4,11,5,2,9,3,8)$ | $\zeta_{12}^{3} - \zeta_{12}$ |
The blue line marks the conjugacy class containing complex conjugation.