Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(2240\)\(\medspace = 2^{6} \cdot 5 \cdot 7 \) |
Artin stem field: | Galois closure of 12.0.629407744000000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.140.6t1.b.b |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.980.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 22x^{4} - 20x^{2} + 9 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{6} + 2x^{4} + 10x^{2} + 3x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 14 a^{5} + 14 a^{4} + 3 a^{3} + 2 a^{2} + 16 a + 10 + \left(3 a^{5} + 6 a^{4} + 15 a^{3} + 6 a^{2} + a + 15\right)\cdot 17 + \left(8 a^{5} + 5 a^{4} + 9 a^{3} + 7 a^{2} + 3 a + 2\right)\cdot 17^{2} + \left(6 a^{5} + a^{4} + 5 a^{3} + 9 a^{2} + 13 a + 11\right)\cdot 17^{3} + \left(15 a^{5} + 10 a^{4} + 8 a^{3} + 11 a^{2} + 4 a + 1\right)\cdot 17^{4} + \left(4 a^{3} + 13 a^{2} + a + 1\right)\cdot 17^{5} + \left(4 a^{5} + 4 a^{4} + 3 a^{3} + 8 a^{2} + 11 a + 3\right)\cdot 17^{6} + \left(8 a^{4} + 10 a^{3} + 6 a^{2} + 3 a + 3\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 2 }$ | $=$ | \( 12 a^{5} + 3 a^{4} + 14 a^{3} + 5 a^{2} + 10 a + 3 + \left(13 a^{5} + 4 a^{4} + 9 a^{3} + 9 a^{2} + 15 a + 12\right)\cdot 17 + \left(2 a^{5} + 10 a^{4} + 16 a^{3} + 8 a^{2} + 5 a + 16\right)\cdot 17^{2} + \left(9 a^{5} + 3 a^{4} + a^{3} + 7 a^{2} + 6 a + 5\right)\cdot 17^{3} + \left(3 a^{5} + 8 a^{4} + a^{3} + 5 a^{2} + 14 a + 4\right)\cdot 17^{4} + \left(8 a^{5} + 15 a^{4} + 5 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 17^{5} + \left(6 a^{5} + 10 a^{4} + 6 a^{3} + 11 a^{2} + 13 a\right)\cdot 17^{6} + \left(3 a^{5} + 10 a^{4} + 2 a^{3} + 3 a^{2} + 15 a\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 3 }$ | $=$ | \( 3 a^{5} + 2 a^{4} + 8 a^{2} + 9 a + 14 + \left(13 a^{5} + 14 a^{4} + 4 a^{3} + 9 a^{2} + 14 a + 15\right)\cdot 17 + \left(9 a^{5} + 10 a^{4} + 4 a^{3} + 4 a^{2} + 4 a + 11\right)\cdot 17^{2} + \left(6 a^{5} + 7 a^{4} + 9 a^{3} + a^{2} + 2\right)\cdot 17^{3} + \left(4 a^{5} + 16 a^{4} + 7 a^{3} + 14 a^{2} + 2 a + 5\right)\cdot 17^{4} + \left(12 a^{5} + 9 a^{4} + 3 a^{3} + 13 a^{2} + 9 a + 7\right)\cdot 17^{5} + \left(16 a^{5} + 9 a^{4} + 12 a^{3} + 12 a^{2} + 11 a + 8\right)\cdot 17^{6} + \left(15 a^{5} + 7 a^{4} + 4 a^{3} + 10 a^{2} + 4 a + 8\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 4 }$ | $=$ | \( 12 a^{5} + 16 a^{4} + 13 a + 3 + \left(3 a^{5} + a^{4} + 10 a^{3} + 4 a^{2} + 16 a + 16\right)\cdot 17 + \left(13 a^{5} + 10 a^{4} + 6 a^{3} + 10 a^{2} + 9 a + 16\right)\cdot 17^{2} + \left(14 a^{5} + 13 a^{4} + 14 a^{3} + 3 a^{2} + 14 a + 16\right)\cdot 17^{3} + \left(a^{5} + 3 a^{4} + 4 a^{3} + 9 a^{2} + 14 a + 14\right)\cdot 17^{4} + \left(6 a^{5} + 15 a^{4} + 14 a^{3} + 16 a^{2} + 7 a + 11\right)\cdot 17^{5} + \left(15 a^{5} + 14 a^{4} + 5 a^{3} + 3 a^{2} + 8 a + 6\right)\cdot 17^{6} + \left(14 a^{5} + a^{4} + 7 a^{3} + 11 a^{2} + 5 a + 3\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 5 }$ | $=$ | \( 14 a^{5} + 4 a^{4} + 10 a^{3} + a^{2} + 5 a + 7 + \left(6 a^{5} + 8 a^{3} + 8 a^{2} + 10 a + 3\right)\cdot 17 + \left(8 a^{5} + 6 a^{4} + 2 a^{3} + 14 a^{2} + 13 a + 14\right)\cdot 17^{2} + \left(13 a^{4} + 12 a^{3} + 16 a^{2} + 11 a + 5\right)\cdot 17^{3} + \left(12 a^{5} + 9 a^{4} + 9 a^{3} + 16 a^{2} + 7 a + 12\right)\cdot 17^{4} + \left(8 a^{5} + 12 a^{4} + 3 a^{3} + 8 a^{2} + 3 a + 7\right)\cdot 17^{5} + \left(16 a^{5} + a^{4} + 16 a^{3} + 11 a^{2} + 14 a + 12\right)\cdot 17^{6} + \left(15 a^{5} + 14 a^{4} + 4 a^{3} + 14 a^{2} + 4 a + 2\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 6 }$ | $=$ | \( 8 a^{5} + 12 a^{4} + 16 a^{3} + 7 a^{2} + 15 a + 14 + \left(4 a^{5} + 7 a^{4} + 2 a^{3} + 16 a^{2} + 15 a + 9\right)\cdot 17 + \left(15 a^{3} + a^{2} + 15 a + 7\right)\cdot 17^{2} + \left(6 a^{5} + 3 a^{3} + 4 a^{2} + 15 a + 16\right)\cdot 17^{3} + \left(4 a^{5} + 16 a^{4} + 14 a^{3} + a^{2} + 10 a + 13\right)\cdot 17^{4} + \left(12 a^{5} + 2 a^{4} + 13 a^{3} + a^{2} + 3\right)\cdot 17^{5} + \left(12 a^{5} + 6 a^{3} + 9 a + 16\right)\cdot 17^{6} + \left(11 a^{5} + 6 a^{4} + 15 a^{3} + a^{2} + 10 a + 2\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 7 }$ | $=$ | \( 3 a^{5} + 3 a^{4} + 14 a^{3} + 15 a^{2} + a + 7 + \left(13 a^{5} + 10 a^{4} + a^{3} + 10 a^{2} + 15 a + 1\right)\cdot 17 + \left(8 a^{5} + 11 a^{4} + 7 a^{3} + 9 a^{2} + 13 a + 14\right)\cdot 17^{2} + \left(10 a^{5} + 15 a^{4} + 11 a^{3} + 7 a^{2} + 3 a + 5\right)\cdot 17^{3} + \left(a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + 12 a + 15\right)\cdot 17^{4} + \left(16 a^{5} + 16 a^{4} + 12 a^{3} + 3 a^{2} + 15 a + 15\right)\cdot 17^{5} + \left(12 a^{5} + 12 a^{4} + 13 a^{3} + 8 a^{2} + 5 a + 13\right)\cdot 17^{6} + \left(16 a^{5} + 8 a^{4} + 6 a^{3} + 10 a^{2} + 13 a + 13\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 8 }$ | $=$ | \( 5 a^{5} + 14 a^{4} + 3 a^{3} + 12 a^{2} + 7 a + 14 + \left(3 a^{5} + 12 a^{4} + 7 a^{3} + 7 a^{2} + a + 4\right)\cdot 17 + \left(14 a^{5} + 6 a^{4} + 8 a^{2} + 11 a\right)\cdot 17^{2} + \left(7 a^{5} + 13 a^{4} + 15 a^{3} + 9 a^{2} + 10 a + 11\right)\cdot 17^{3} + \left(13 a^{5} + 8 a^{4} + 15 a^{3} + 11 a^{2} + 2 a + 12\right)\cdot 17^{4} + \left(8 a^{5} + a^{4} + 11 a^{3} + 9 a^{2} + 14 a + 11\right)\cdot 17^{5} + \left(10 a^{5} + 6 a^{4} + 10 a^{3} + 5 a^{2} + 3 a + 16\right)\cdot 17^{6} + \left(13 a^{5} + 6 a^{4} + 14 a^{3} + 13 a^{2} + a + 16\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 9 }$ | $=$ | \( 5 a^{5} + a^{4} + 4 a + 14 + \left(13 a^{5} + 15 a^{4} + 7 a^{3} + 13 a^{2}\right)\cdot 17 + \left(3 a^{5} + 6 a^{4} + 10 a^{3} + 6 a^{2} + 7 a\right)\cdot 17^{2} + \left(2 a^{5} + 3 a^{4} + 2 a^{3} + 13 a^{2} + 2 a\right)\cdot 17^{3} + \left(15 a^{5} + 13 a^{4} + 12 a^{3} + 7 a^{2} + 2 a + 2\right)\cdot 17^{4} + \left(10 a^{5} + a^{4} + 2 a^{3} + 9 a + 5\right)\cdot 17^{5} + \left(a^{5} + 2 a^{4} + 11 a^{3} + 13 a^{2} + 8 a + 10\right)\cdot 17^{6} + \left(2 a^{5} + 15 a^{4} + 9 a^{3} + 5 a^{2} + 11 a + 13\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 10 }$ | $=$ | \( 14 a^{5} + 15 a^{4} + 9 a^{2} + 8 a + 3 + \left(3 a^{5} + 2 a^{4} + 13 a^{3} + 7 a^{2} + 2 a + 1\right)\cdot 17 + \left(7 a^{5} + 6 a^{4} + 12 a^{3} + 12 a^{2} + 12 a + 5\right)\cdot 17^{2} + \left(10 a^{5} + 9 a^{4} + 7 a^{3} + 15 a^{2} + 16 a + 14\right)\cdot 17^{3} + \left(12 a^{5} + 9 a^{3} + 2 a^{2} + 14 a + 11\right)\cdot 17^{4} + \left(4 a^{5} + 7 a^{4} + 13 a^{3} + 3 a^{2} + 7 a + 9\right)\cdot 17^{5} + \left(7 a^{4} + 4 a^{3} + 4 a^{2} + 5 a + 8\right)\cdot 17^{6} + \left(a^{5} + 9 a^{4} + 12 a^{3} + 6 a^{2} + 12 a + 8\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 11 }$ | $=$ | \( 3 a^{5} + 13 a^{4} + 7 a^{3} + 16 a^{2} + 12 a + 10 + \left(10 a^{5} + 16 a^{4} + 8 a^{3} + 8 a^{2} + 6 a + 13\right)\cdot 17 + \left(8 a^{5} + 10 a^{4} + 14 a^{3} + 2 a^{2} + 3 a + 2\right)\cdot 17^{2} + \left(16 a^{5} + 3 a^{4} + 4 a^{3} + 5 a + 11\right)\cdot 17^{3} + \left(4 a^{5} + 7 a^{4} + 7 a^{3} + 9 a + 4\right)\cdot 17^{4} + \left(8 a^{5} + 4 a^{4} + 13 a^{3} + 8 a^{2} + 13 a + 9\right)\cdot 17^{5} + \left(15 a^{4} + 5 a^{2} + 2 a + 4\right)\cdot 17^{6} + \left(a^{5} + 2 a^{4} + 12 a^{3} + 2 a^{2} + 12 a + 14\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 12 }$ | $=$ | \( 9 a^{5} + 5 a^{4} + a^{3} + 10 a^{2} + 2 a + 3 + \left(12 a^{5} + 9 a^{4} + 14 a^{3} + a + 7\right)\cdot 17 + \left(16 a^{5} + 16 a^{4} + a^{3} + 15 a^{2} + a + 9\right)\cdot 17^{2} + \left(10 a^{5} + 16 a^{4} + 13 a^{3} + 12 a^{2} + a\right)\cdot 17^{3} + \left(12 a^{5} + 2 a^{3} + 15 a^{2} + 6 a + 3\right)\cdot 17^{4} + \left(4 a^{5} + 14 a^{4} + 3 a^{3} + 15 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(4 a^{5} + 16 a^{4} + 10 a^{3} + 16 a^{2} + 7 a\right)\cdot 17^{6} + \left(5 a^{5} + 10 a^{4} + a^{3} + 15 a^{2} + 6 a + 14\right)\cdot 17^{7} +O(17^{8})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,8)(3,10)(4,9)(5,11)(6,12)$ | $-2$ |
$3$ | $2$ | $(1,6)(2,4)(3,5)(7,12)(8,9)(10,11)$ | $0$ |
$3$ | $2$ | $(1,8)(2,7)(3,9)(4,10)(5,12)(6,11)$ | $0$ |
$1$ | $3$ | $(1,5,4)(2,6,3)(7,11,9)(8,12,10)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,4,5)(2,3,6)(7,9,11)(8,10,12)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(2,6,3)(8,12,10)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(2,3,6)(8,10,12)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,4,5)(2,6,3)(7,9,11)(8,12,10)$ | $-1$ |
$1$ | $6$ | $(1,9,5,7,4,11)(2,10,6,8,3,12)$ | $2 \zeta_{3} + 2$ |
$1$ | $6$ | $(1,11,4,7,5,9)(2,12,3,8,6,10)$ | $-2 \zeta_{3}$ |
$2$ | $6$ | $(1,7)(2,10,6,8,3,12)(4,9)(5,11)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,7)(2,12,3,8,6,10)(4,9)(5,11)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,11,4,7,5,9)(2,10,6,8,3,12)$ | $1$ |
$3$ | $6$ | $(1,12,4,8,5,10)(2,11,3,7,6,9)$ | $0$ |
$3$ | $6$ | $(1,10,5,8,4,12)(2,9,6,7,3,11)$ | $0$ |
$3$ | $6$ | $(1,2,5,6,4,3)(7,8,11,12,9,10)$ | $0$ |
$3$ | $6$ | $(1,3,4,6,5,2)(7,10,9,12,11,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.