Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \) |
Artin stem field: | Galois closure of 12.0.4822937894390625.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.855.6t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.7695.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 5 x^{11} + 16 x^{10} - 27 x^{9} + 12 x^{8} + 45 x^{7} - 121 x^{6} + 90 x^{5} + 87 x^{4} + \cdots + 44 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{6} + x^{4} + 9x^{3} + 9x^{2} + x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( a^{5} + 15 a^{4} + 5 a^{3} + 4 a^{2} + 6 a + 5 + \left(5 a^{5} + 6 a^{4} + 7 a^{3} + 22 a^{2} + 6 a + 3\right)\cdot 23 + \left(2 a^{5} + 9 a^{4} + 15 a^{3} + 14 a + 7\right)\cdot 23^{2} + \left(22 a^{5} + 11 a^{4} + 16 a^{3} + 10 a^{2} + 4 a + 19\right)\cdot 23^{3} + \left(2 a^{5} + 16 a^{4} + a^{3} + 22 a^{2} + 12 a + 19\right)\cdot 23^{4} + \left(19 a^{5} + 18 a^{4} + 5 a^{3} + 2 a^{2} + 16 a + 4\right)\cdot 23^{5} + \left(13 a^{5} + 15 a^{4} + 8 a^{3} + 12 a^{2} + 18 a + 2\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 2 }$ | $=$ | \( 12 a^{5} + 3 a^{4} + a^{3} + 3 a^{2} + 18 a + 14 + \left(21 a^{5} + 4 a^{4} + 3 a^{3} + 3 a^{2} + 13 a + 22\right)\cdot 23 + \left(14 a^{5} + 14 a^{4} + a^{3} + 16 a^{2} + 11 a + 13\right)\cdot 23^{2} + \left(6 a^{5} + 4 a^{4} + 11 a^{3} + 22 a^{2} + 20 a + 20\right)\cdot 23^{3} + \left(14 a^{4} + 11 a^{3} + 5 a^{2} + 9 a + 20\right)\cdot 23^{4} + \left(3 a^{5} + 16 a^{4} + 8 a^{3} + 16 a^{2} + 21 a + 12\right)\cdot 23^{5} + \left(17 a^{5} + 11 a^{4} + 15 a^{3} + 3 a^{2} + 8 a + 5\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 3 }$ | $=$ | \( 13 a^{5} + 9 a^{4} + 6 a^{3} + 14 a^{2} + 17 a + 2 + \left(5 a^{4} + 16 a^{3} + 12 a^{2} + 12 a + 3\right)\cdot 23 + \left(18 a^{5} + 12 a^{4} + 14 a^{2} + 2 a + 11\right)\cdot 23^{2} + \left(10 a^{5} + 9 a^{4} + 6 a^{2} + 15 a + 14\right)\cdot 23^{3} + \left(19 a^{5} + 17 a^{4} + 3 a^{2} + 13 a + 4\right)\cdot 23^{4} + \left(3 a^{5} + 13 a^{4} + 6 a^{2} + 14 a + 21\right)\cdot 23^{5} + \left(8 a^{5} + a^{4} + 7 a^{3} + 10 a^{2} + 3 a + 21\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 4 }$ | $=$ | \( 20 a^{5} + 4 a^{4} + 22 a^{3} + a^{2} + 7 a + 10 + \left(8 a^{5} + a^{4} + 5 a^{3} + 13 a^{2} + 20 a + 19\right)\cdot 23 + \left(5 a^{4} + 10 a^{3} + 20 a^{2} + 4 a + 21\right)\cdot 23^{2} + \left(12 a^{5} + 16 a^{4} + 22 a^{3} + a^{2} + 4 a + 16\right)\cdot 23^{3} + \left(15 a^{5} + 3 a^{4} + 15 a^{3} + 9 a^{2} + 19 a + 11\right)\cdot 23^{4} + \left(13 a^{5} + 12 a^{4} + 21 a^{3} + 20 a^{2} + 10 a + 14\right)\cdot 23^{5} + \left(15 a^{5} + 13 a^{4} + 16 a^{3} + 8 a + 12\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 5 }$ | $=$ | \( 18 a^{5} + 22 a^{4} + 17 a^{3} + 5 a^{2} + 1 + \left(16 a^{5} + 21 a^{4} + 7 a^{3} + 4 a^{2} + 5 a + 8\right)\cdot 23 + \left(22 a^{5} + 2 a^{4} + 11 a^{3} + 14 a^{2} + 17 a + 20\right)\cdot 23^{2} + \left(20 a^{4} + 11 a^{3} + 16 a^{2} + 20 a + 15\right)\cdot 23^{3} + \left(20 a^{5} + 20 a^{4} + 21 a^{3} + 16 a^{2} + 2 a + 21\right)\cdot 23^{4} + \left(16 a^{5} + 21 a^{4} + 6 a^{3} + 5 a^{2} + a + 8\right)\cdot 23^{5} + \left(22 a^{5} + 9 a^{4} + 18 a^{3} + 17 a^{2} + 14 a + 10\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 6 }$ | $=$ | \( a^{5} + 8 a^{4} + 3 a^{3} + 11 a^{2} + 15 a + 12 + \left(5 a^{5} + 15 a^{4} + 15 a^{3} + 18 a^{2} + 11 a + 3\right)\cdot 23 + \left(10 a^{5} + 8 a^{4} + 6 a^{3} + 2 a^{2} + 14 a + 12\right)\cdot 23^{2} + \left(14 a^{5} + 2 a^{4} + 13 a^{3} + 14 a^{2} + 5 a + 15\right)\cdot 23^{3} + \left(2 a^{5} + 8 a^{4} + 8 a^{3} + 17 a^{2} + 15 a + 11\right)\cdot 23^{4} + \left(3 a^{5} + 4 a^{4} + 2 a^{2} + 4 a + 15\right)\cdot 23^{5} + \left(2 a^{5} + 14 a^{4} + 8 a^{2} + 1\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{5} + 12 a^{4} + 16 a^{3} + 20 a^{2} + 13 a + 6 + \left(7 a^{4} + 21 a^{3} + 5 a^{2} + 17 a + 9\right)\cdot 23 + \left(11 a^{4} + 19 a^{3} + 22 a^{2} + 8 a + 11\right)\cdot 23^{2} + \left(13 a^{5} + 5 a^{4} + 10 a^{3} + 7 a^{2} + 20 a\right)\cdot 23^{3} + \left(21 a^{5} + 17 a^{4} + 18 a^{3} + 12 a^{2} + 2 a + 18\right)\cdot 23^{4} + \left(16 a^{5} + 6 a^{4} + a^{3} + 4 a^{2} + 6 a + 21\right)\cdot 23^{5} + \left(14 a^{5} + 10 a^{4} + 8 a^{3} + 22 a^{2} + 5 a + 5\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{5} + 20 a^{4} + 22 a^{3} + 13 a^{2} + 5 a + 5 + \left(19 a^{5} + 18 a^{4} + 21 a^{3} + 5 a^{2} + 13 a + 2\right)\cdot 23 + \left(8 a^{5} + 5 a^{4} + 3 a^{3} + 9 a + 3\right)\cdot 23^{2} + \left(9 a^{5} + 10 a^{4} + 4 a^{3} + 21 a^{2} + 9 a + 18\right)\cdot 23^{3} + \left(2 a^{5} + 9 a^{4} + 7 a^{3} + 19 a^{2} + 10 a + 18\right)\cdot 23^{4} + \left(15 a^{5} + 7 a^{4} + 13 a^{3} + 10 a^{2} + 16 a + 18\right)\cdot 23^{5} + \left(4 a^{5} + 3 a^{4} + 7 a^{3} + 10 a^{2} + 9 a + 19\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 9 }$ | $=$ | \( 20 a^{5} + 14 a^{4} + 17 a^{3} + 8 a^{2} + 6 a + 12 + \left(13 a^{5} + 6 a^{4} + 20 a^{3} + 13 a^{2} + 15 a + 14\right)\cdot 23 + \left(20 a^{5} + 20 a^{4} + 3 a^{3} + 11 a^{2} + 14 a + 14\right)\cdot 23^{2} + \left(18 a^{5} + 13 a^{4} + 10 a^{3} + 21 a^{2} + 11 a + 12\right)\cdot 23^{3} + \left(11 a^{5} + 18 a^{4} + 19 a^{3} + 19 a^{2} + 13 a + 13\right)\cdot 23^{4} + \left(21 a^{5} + a^{3} + 14 a^{2} + 22 a + 17\right)\cdot 23^{5} + \left(7 a^{4} + 20 a^{3} + 5 a^{2} + 16 a + 8\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 10 }$ | $=$ | \( 4 a^{5} + 2 a^{4} + 21 a^{3} + 13 a^{2} + 10 a + 9 + \left(5 a^{5} + 11 a^{4} + 21 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 23 + \left(16 a^{5} + 15 a^{4} + 13 a^{3} + 17 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(18 a^{5} + a^{4} + 9 a^{3} + 9 a^{2} + 5 a + 14\right)\cdot 23^{3} + \left(22 a^{4} + 12 a^{3} + 4 a^{2} + 13 a + 4\right)\cdot 23^{4} + \left(21 a^{5} + 15 a^{4} + 17 a^{3} + 7 a^{2} + 2\right)\cdot 23^{5} + \left(18 a^{5} + 8 a^{4} + 10 a + 5\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 11 }$ | $=$ | \( 12 a^{5} + 20 a^{4} + 8 a^{3} + a^{2} + 4 a + 7 + \left(5 a^{5} + 9 a^{4} + 7 a^{3} + 5 a^{2} + 13 a + 9\right)\cdot 23 + \left(20 a^{5} + 2 a^{4} + 17 a^{3} + 15 a^{2} + 7 a + 11\right)\cdot 23^{2} + \left(16 a^{5} + 17 a^{4} + 14 a^{3} + 13 a^{2} + 16 a + 21\right)\cdot 23^{3} + \left(21 a^{5} + 13 a^{4} + 8 a^{3} + 14 a^{2} + 7 a + 14\right)\cdot 23^{4} + \left(20 a^{5} + 20 a^{4} + 10 a^{3} + 5 a^{2} + 7 a + 16\right)\cdot 23^{5} + \left(18 a^{5} + 17 a^{4} + 11 a^{3} + 3 a^{2} + 12 a + 14\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 12 }$ | $=$ | \( 11 a^{5} + 9 a^{4} + 22 a^{2} + 14 a + 14 + \left(12 a^{5} + 6 a^{4} + 12 a^{3} + 22 a^{2} + a + 5\right)\cdot 23 + \left(3 a^{5} + 7 a^{4} + 10 a^{3} + a^{2} + 13 a + 12\right)\cdot 23^{2} + \left(17 a^{5} + 2 a^{4} + 13 a^{3} + 15 a^{2} + 3 a + 13\right)\cdot 23^{3} + \left(18 a^{5} + 22 a^{4} + 12 a^{3} + 14 a^{2} + 17 a\right)\cdot 23^{4} + \left(5 a^{5} + 21 a^{4} + 4 a^{3} + 17 a^{2} + 15 a + 6\right)\cdot 23^{5} + \left(a^{3} + 20 a^{2} + 6 a + 6\right)\cdot 23^{6} +O(23^{7})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)(5,6)(7,9)(8,10)(11,12)$ | $-2$ |
$3$ | $2$ | $(1,11)(2,5)(3,12)(4,6)(7,8)(9,10)$ | $0$ |
$3$ | $2$ | $(1,12)(2,6)(3,11)(4,5)(7,10)(8,9)$ | $0$ |
$1$ | $3$ | $(1,8,5)(2,11,7)(3,10,6)(4,12,9)$ | $-2 \zeta_{3} - 2$ |
$1$ | $3$ | $(1,5,8)(2,7,11)(3,6,10)(4,9,12)$ | $2 \zeta_{3}$ |
$2$ | $3$ | $(1,5,8)(2,11,7)(3,6,10)(4,12,9)$ | $-1$ |
$2$ | $3$ | $(2,7,11)(4,9,12)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(2,11,7)(4,12,9)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,10,5,3,8,6)(2,12,7,4,11,9)$ | $2 \zeta_{3} + 2$ |
$1$ | $6$ | $(1,6,8,3,5,10)(2,9,11,4,7,12)$ | $-2 \zeta_{3}$ |
$2$ | $6$ | $(1,10,5,3,8,6)(2,9,11,4,7,12)$ | $1$ |
$2$ | $6$ | $(1,3)(2,9,11,4,7,12)(5,6)(8,10)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,3)(2,12,7,4,11,9)(5,6)(8,10)$ | $\zeta_{3}$ |
$3$ | $6$ | $(1,2,8,11,5,7)(3,4,10,12,6,9)$ | $0$ |
$3$ | $6$ | $(1,7,5,11,8,2)(3,9,6,12,10,4)$ | $0$ |
$3$ | $6$ | $(1,4,8,12,5,9)(2,10,11,6,7,3)$ | $0$ |
$3$ | $6$ | $(1,9,5,12,8,4)(2,3,7,6,11,10)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.