Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Artin stem field: | Galois closure of 12.0.3087580356000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.315.6t1.h.b |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.11340.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} + 4x^{10} - x^{9} - 6x^{8} + x^{7} - 35x^{6} - 22x^{5} + 32x^{4} - 42x^{3} + 128x^{2} + 156x + 116 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{6} + 2x^{4} + 10x^{2} + 3x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 16 + \left(7 a^{5} + 3 a^{4} + 2 a^{3} + 6 a^{2} + 15 a + 2\right)\cdot 17 + \left(11 a^{5} + 2 a^{4} + 13 a^{3} + 5 a^{2} + 12 a + 13\right)\cdot 17^{2} + \left(2 a^{5} + 4 a^{4} + 9 a^{3} + a^{2} + 7 a + 9\right)\cdot 17^{3} + \left(4 a^{5} + 4 a^{4} + 3 a^{3} + 6 a + 11\right)\cdot 17^{4} + \left(16 a^{5} + 10 a^{4} + 9 a^{3} + 7 a^{2} + 3 a + 9\right)\cdot 17^{5} + \left(14 a^{5} + 10 a^{3} + 3 a^{2} + 9 a + 14\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 2 }$ | $=$ | \( 12 a^{5} + 13 a^{4} + 3 a^{3} + 7 a^{2} + 5 a + 6 + \left(2 a^{5} + 6 a^{4} + 12 a^{3} + 3 a^{2} + 14 a + 6\right)\cdot 17 + \left(14 a^{5} + 9 a^{4} + 4 a^{3} + 3 a^{2} + 10 a\right)\cdot 17^{2} + \left(8 a^{5} + 10 a^{4} + 2 a^{3} + 14 a^{2} + 15 a\right)\cdot 17^{3} + \left(9 a^{5} + 13 a^{4} + 9 a^{3} + 5 a^{2} + a + 8\right)\cdot 17^{4} + \left(a^{5} + 9 a^{4} + 14 a^{3} + a^{2} + a + 6\right)\cdot 17^{5} + \left(14 a^{5} + 12 a^{4} + 13 a^{3} + 5 a^{2} + a + 9\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 3 }$ | $=$ | \( 13 a^{5} + 13 a^{3} + 11 a^{2} + 2 a + 7 + \left(2 a^{4} + 9 a^{3} + 4 a^{2} + 3 a + 2\right)\cdot 17 + \left(12 a^{5} + 4 a^{4} + 15 a^{3} + 16 a^{2} + 1\right)\cdot 17^{2} + \left(6 a^{5} + 11 a^{4} + 15 a^{3} + 3 a^{2} + 9 a\right)\cdot 17^{3} + \left(14 a^{5} + 15 a^{4} + 6 a^{2} + 3 a + 3\right)\cdot 17^{4} + \left(3 a^{5} + 11 a^{4} + 14 a^{3} + 3 a^{2} + 3 a + 2\right)\cdot 17^{5} + \left(12 a^{5} + 16 a^{4} + 11 a^{3} + 7 a + 12\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{5} + a^{4} + 9 a^{3} + 15 a^{2} + 13 a + 5 + \left(13 a^{5} + 10 a^{4} + 6 a^{3} + 9 a^{2} + 14 a + 11\right)\cdot 17 + \left(15 a^{5} + 7 a^{4} + 4 a^{3} + 2 a^{2} + 9 a + 4\right)\cdot 17^{2} + \left(13 a^{5} + 15 a^{4} + 9 a^{3} + 14 a^{2} + 7 a + 13\right)\cdot 17^{3} + \left(3 a^{5} + 3 a^{4} + 11 a^{3} + 10 a^{2} + 14 a + 13\right)\cdot 17^{4} + \left(12 a^{5} + 3 a^{4} + 14 a^{3} + 13 a^{2} + 10 a + 16\right)\cdot 17^{5} + \left(10 a^{5} + 14 a^{4} + 13 a^{2} + 4 a + 3\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 5 }$ | $=$ | \( a^{5} + 3 a^{4} + 2 a^{3} + 8 a^{2} + 7 a + 3 + \left(9 a^{5} + 2 a^{4} + a^{3} + 8 a^{2} + 6 a + 2\right)\cdot 17 + \left(16 a^{5} + 5 a^{4} + a^{3} + 10 a^{2} + 2 a + 7\right)\cdot 17^{2} + \left(2 a^{5} + 15 a^{4} + 12 a^{3} + a^{2} + a + 16\right)\cdot 17^{3} + \left(11 a^{5} + 6 a^{4} + 2 a^{3} + 3 a^{2} + a + 8\right)\cdot 17^{4} + \left(6 a^{5} + 11 a^{4} + 11 a^{3} + 6 a^{2} + 2 a + 7\right)\cdot 17^{5} + \left(8 a^{5} + 3 a^{4} + 3 a^{3} + 3 a^{2} + 11 a + 14\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 6 }$ | $=$ | \( 9 a^{5} + 13 a^{4} + 9 a^{3} + 13 a^{2} + 14 a + 6 + \left(7 a^{5} + 10 a^{4} + 10 a^{3} + 9 a^{2} + 3 a + 5\right)\cdot 17 + \left(7 a^{5} + 15 a^{4} + 11 a^{3} + 14 a^{2} + 7 a + 6\right)\cdot 17^{2} + \left(3 a^{5} + 14 a^{3} + 7 a^{2} + a + 7\right)\cdot 17^{3} + \left(14 a^{5} + 5 a^{4} + 12 a^{3} + 4 a^{2} + 9 a + 10\right)\cdot 17^{4} + \left(5 a^{5} + 16 a^{4} + 6 a^{3} + 13 a^{2} + 15 a + 13\right)\cdot 17^{5} + \left(3 a^{5} + a^{4} + 6 a^{3} + 10 a^{2} + 10 a + 11\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 7 }$ | $=$ | \( 6 a^{5} + 4 a^{4} + 3 a^{3} + 6 a^{2} + 10 a + 16 + \left(11 a^{5} + 12 a^{4} + 3 a^{3} + 2 a^{2} + 10 a + 13\right)\cdot 17 + \left(14 a^{5} + 8 a^{4} + 2 a^{3} + 2 a + 15\right)\cdot 17^{2} + \left(a^{5} + 9 a^{4} + 8 a^{3} + 11 a^{2} + 3 a + 8\right)\cdot 17^{3} + \left(8 a^{5} + 4 a^{4} + 11 a^{3} + 14 a^{2} + 4 a + 4\right)\cdot 17^{4} + \left(7 a^{5} + 6 a^{3} + 4 a^{2} + 15 a + 15\right)\cdot 17^{5} + \left(4 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + 8 a + 1\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 8 }$ | $=$ | \( 2 a^{5} + 9 a^{3} + 14 a^{2} + 2 a + 7 + \left(5 a^{5} + 11 a^{4} + 10 a^{3} + 15 a + 13\right)\cdot 17 + \left(15 a^{5} + 10 a^{4} + 4 a^{3} + 11 a^{2} + 6 a + 3\right)\cdot 17^{2} + \left(5 a^{5} + 16 a^{4} + 14 a^{3} + 13 a^{2} + 4 a + 16\right)\cdot 17^{3} + \left(a^{5} + 15 a^{4} + 8 a^{3} + 3 a^{2} + a + 12\right)\cdot 17^{4} + \left(12 a^{5} + a^{4} + 11 a^{3} + a^{2} + 15 a + 7\right)\cdot 17^{5} + \left(a^{5} + 8 a^{4} + 4 a^{3} + 6 a + 16\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 9 }$ | $=$ | \( a^{5} + 13 a^{4} + 16 a^{3} + 10 a^{2} + 5 a + 6 + \left(7 a^{5} + 15 a^{4} + 12 a^{3} + 16 a^{2} + 9 a\right)\cdot 17 + \left(15 a^{4} + 10 a^{3} + 14 a^{2} + 3\right)\cdot 17^{2} + \left(8 a^{5} + 15 a^{4} + 6 a^{2} + 11 a + 16\right)\cdot 17^{3} + \left(13 a^{5} + 13 a^{4} + 3 a^{2} + 16 a\right)\cdot 17^{4} + \left(9 a^{5} + 16 a^{4} + 12 a^{3} + 16 a^{2} + 12 a + 12\right)\cdot 17^{5} + \left(3 a^{5} + 3 a^{4} + 6 a^{3} + 4 a^{2} + 13\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 10 }$ | $=$ | \( 14 a^{5} + a^{4} + 7 a^{3} + 8 a^{2} + 13 a + 5 + \left(4 a^{5} + 6 a^{4} + 11 a^{3} + 2 a^{2} + 3 a + 8\right)\cdot 17 + \left(6 a^{5} + 11 a^{4} + 15 a^{3} + a^{2} + 8 a + 8\right)\cdot 17^{2} + \left(5 a^{4} + 7 a^{3} + 4 a^{2} + 6 a + 5\right)\cdot 17^{3} + \left(7 a^{5} + 10 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 6\right)\cdot 17^{4} + \left(a^{5} + 16 a^{4} + 15 a^{3} + 14 a^{2} + 14 a + 13\right)\cdot 17^{5} + \left(7 a^{5} + 4 a^{3} + 3 a\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 11 }$ | $=$ | \( 4 a^{5} + 3 a^{4} + a^{2} + 7 a + 3 + \left(15 a^{4} + 6 a^{3} + a^{2} + 12 a + 16\right)\cdot 17 + \left(7 a^{5} + 8 a^{4} + 12 a^{3} + 9 a^{2} + 10\right)\cdot 17^{2} + \left(6 a^{5} + 5 a^{4} + 10 a^{3} + 8 a^{2} + 8\right)\cdot 17^{3} + \left(14 a^{5} + 13 a^{4} + a^{3} + 13 a^{2} + 13 a + 1\right)\cdot 17^{4} + \left(12 a^{5} + 7 a^{4} + 12 a^{3} + 6 a^{2} + 5 a + 4\right)\cdot 17^{5} + \left(4 a^{5} + 7 a^{4} + 7 a^{3} + 7 a^{2} + 10 a + 11\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 12 }$ | $=$ | \( 12 a^{5} + 13 a^{4} + 7 a^{3} + 6 a^{2} + 14 a + 6 + \left(15 a^{5} + 6 a^{4} + 15 a^{3} + 2 a^{2} + 9 a + 2\right)\cdot 17 + \left(14 a^{5} + 2 a^{4} + 5 a^{3} + 13 a^{2} + 5 a + 10\right)\cdot 17^{2} + \left(6 a^{5} + 8 a^{4} + 13 a^{3} + 14 a^{2} + 16\right)\cdot 17^{3} + \left(11 a^{4} + 11 a^{3} + 14 a^{2} + 4 a + 2\right)\cdot 17^{4} + \left(12 a^{5} + 12 a^{4} + 7 a^{3} + 13 a^{2} + 2 a + 10\right)\cdot 17^{5} + \left(16 a^{5} + 5 a^{4} + 10 a^{3} + 14 a^{2} + 10 a + 8\right)\cdot 17^{6} +O(17^{7})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,9)(3,8)(4,10)(5,11)(6,12)$ | $-2$ |
$3$ | $2$ | $(1,5)(2,4)(3,6)(7,11)(8,12)(9,10)$ | $0$ |
$3$ | $2$ | $(1,11)(2,10)(3,12)(4,9)(5,7)(6,8)$ | $0$ |
$1$ | $3$ | $(1,3,2)(4,5,6)(7,8,9)(10,11,12)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)(10,12,11)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(1,3,2)(4,6,5)(7,8,9)(10,12,11)$ | $-1$ |
$2$ | $3$ | $(4,6,5)(10,12,11)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(4,5,6)(10,11,12)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,8,2,7,3,9)(4,11,6,10,5,12)$ | $-2 \zeta_{3}$ |
$1$ | $6$ | $(1,9,3,7,2,8)(4,12,5,10,6,11)$ | $2 \zeta_{3} + 2$ |
$2$ | $6$ | $(1,8,2,7,3,9)(4,12,5,10,6,11)$ | $1$ |
$2$ | $6$ | $(1,7)(2,9)(3,8)(4,12,5,10,6,11)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,7)(2,9)(3,8)(4,11,6,10,5,12)$ | $-\zeta_{3} - 1$ |
$3$ | $6$ | $(1,5,2,4,3,6)(7,11,9,10,8,12)$ | $0$ |
$3$ | $6$ | $(1,6,3,4,2,5)(7,12,8,10,9,11)$ | $0$ |
$3$ | $6$ | $(1,11,2,10,3,12)(4,8,6,7,5,9)$ | $0$ |
$3$ | $6$ | $(1,12,3,10,2,11)(4,9,5,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.