Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(3775\)\(\medspace = 5^{2} \cdot 151 \) |
Artin stem field: | Galois closure of 12.0.1015401564453125.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3 \times C_4$ |
Parity: | odd |
Determinant: | 1.151.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.755.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} + 5x^{10} + 10x^{9} - 9x^{8} + 44x^{6} + 100x^{5} + 81x^{4} - 40x^{3} - 40x^{2} + 24x + 16 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{4} + 7x^{2} + 10x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{3} + 14 a^{2} + 8 a + 6 + \left(13 a^{3} + 16 a^{2} + 11 a + 14\right)\cdot 17 + \left(5 a^{3} + 13 a^{2} + 9 a\right)\cdot 17^{2} + \left(12 a^{3} + 13 a^{2} + 14 a + 14\right)\cdot 17^{3} + \left(2 a^{2} + 2\right)\cdot 17^{4} + \left(8 a^{3} + 9 a^{2} + 9 a + 5\right)\cdot 17^{5} + \left(14 a^{3} + 13 a\right)\cdot 17^{6} + \left(10 a^{3} + 15 a^{2} + 14 a + 4\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 2 }$ | $=$ | \( 4 a^{3} + 16 a^{2} + 8 a + 13 + \left(9 a^{3} + a^{2} + 2 a + 7\right)\cdot 17 + \left(16 a^{3} + 8 a^{2} + 6 a + 14\right)\cdot 17^{2} + \left(3 a^{3} + 12 a^{2} + a + 16\right)\cdot 17^{3} + \left(16 a^{2} + 11 a + 9\right)\cdot 17^{4} + \left(11 a^{2} + 4 a + 4\right)\cdot 17^{5} + \left(2 a^{2} + 2 a + 1\right)\cdot 17^{6} + \left(9 a^{3} + 6 a^{2} + 16 a + 1\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 3 }$ | $=$ | \( a^{3} + 15 a^{2} + 13 a + 13 + \left(8 a^{3} + 5 a^{2} + 5 a\right)\cdot 17 + \left(2 a^{3} + 15 a^{2} + 2 a\right)\cdot 17^{2} + \left(3 a^{3} + 8 a^{2} + 8 a + 13\right)\cdot 17^{3} + \left(a^{2} + 14 a + 5\right)\cdot 17^{4} + \left(9 a^{3} + a^{2} + 14 a + 13\right)\cdot 17^{5} + \left(5 a^{3} + 3 a^{2} + 13 a + 4\right)\cdot 17^{6} + \left(12 a^{2} + 16 a + 12\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{3} + 8 a^{2} + 8 a + 13 + \left(9 a^{3} + 7 a^{2} + 5 a + 5\right)\cdot 17 + \left(3 a^{3} + 13 a^{2} + 9 a + 1\right)\cdot 17^{2} + \left(8 a^{3} + 4 a^{2} + 2 a + 11\right)\cdot 17^{3} + \left(6 a^{3} + 4 a^{2} + 8 a + 11\right)\cdot 17^{4} + \left(6 a^{3} + 16 a^{2} + 12 a + 12\right)\cdot 17^{5} + \left(3 a^{2} + 6 a + 11\right)\cdot 17^{6} + \left(6 a^{3} + 2 a + 4\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 5 }$ | $=$ | \( 15 a^{3} + 8 a^{2} + a + 8 + \left(9 a^{3} + a^{2} + 2 a + 2\right)\cdot 17 + \left(6 a^{3} + a^{2} + 12 a\right)\cdot 17^{2} + \left(10 a^{3} + 9 a^{2} + 16 a + 2\right)\cdot 17^{3} + \left(2 a^{3} + 12 a^{2} + 15 a + 13\right)\cdot 17^{4} + \left(5 a^{3} + 2 a^{2} + 6 a + 10\right)\cdot 17^{5} + \left(6 a^{3} + 14 a^{2} + 4 a + 3\right)\cdot 17^{6} + \left(2 a^{3} + 4 a + 9\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 6 }$ | $=$ | \( 9 a^{3} + 16 a^{2} + 13 a + 10 + \left(3 a^{2} + 4 a\right)\cdot 17 + \left(10 a^{3} + 7 a + 10\right)\cdot 17^{2} + \left(12 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 17^{3} + \left(9 a^{3} + 13 a^{2} + 10 a + 14\right)\cdot 17^{4} + \left(7 a^{3} + 13 a^{2} + 11 a\right)\cdot 17^{5} + \left(5 a^{3} + 16 a^{2} + 15\right)\cdot 17^{6} + \left(5 a^{3} + 8 a^{2} + 13 a\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 7 }$ | $=$ | \( 11 a^{3} + 2 a^{2} + 5 a + 10 + \left(5 a^{3} + 3 a^{2} + 8 a + 10\right)\cdot 17 + \left(8 a^{3} + 16 a + 5\right)\cdot 17^{2} + \left(16 a^{3} + 7 a^{2} + 5 a + 4\right)\cdot 17^{3} + \left(13 a^{3} + 8 a^{2} + 4 a + 11\right)\cdot 17^{4} + \left(12 a^{3} + 9 a^{2} + 10 a + 16\right)\cdot 17^{5} + \left(15 a^{3} + 9 a^{2} + 8 a + 15\right)\cdot 17^{6} + \left(7 a^{3} + 10 a^{2} + 13 a + 8\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 8 }$ | $=$ | \( 9 a^{2} + a + 1 + \left(16 a^{3} + 11 a^{2} + 12 a + 7\right)\cdot 17 + \left(16 a^{3} + 15 a^{2} + 6 a + 1\right)\cdot 17^{2} + \left(a^{3} + 11 a^{2} + 11 a\right)\cdot 17^{3} + \left(10 a^{3} + 5 a^{2} + 14 a + 3\right)\cdot 17^{4} + \left(16 a^{3} + 10 a^{2} + 12 a + 4\right)\cdot 17^{5} + \left(13 a^{3} + a^{2}\right)\cdot 17^{6} + \left(12 a^{3} + 12 a\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 9 }$ | $=$ | \( 15 a^{3} + a^{2} + 7 a + 9 + \left(15 a^{3} + 2 a^{2} + 15\right)\cdot 17 + \left(10 a^{3} + 9 a^{2} + 9 a + 9\right)\cdot 17^{2} + \left(4 a + 9\right)\cdot 17^{3} + \left(4 a^{3} + 16 a^{2} + 9 a + 10\right)\cdot 17^{4} + \left(12 a^{3} + 8 a^{2} + 9 a\right)\cdot 17^{5} + \left(13 a^{3} + 15 a^{2} + 9 a + 13\right)\cdot 17^{6} + \left(9 a^{3} + 9 a^{2} + a + 11\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 10 }$ | $=$ | \( 8 a^{3} + 2 a^{2} + 8 a + 13 + \left(14 a^{3} + 10 a^{2} + 9 a + 7\right)\cdot 17 + \left(9 a^{3} + 2 a^{2} + 8\right)\cdot 17^{2} + \left(9 a^{3} + 2 a^{2} + 6 a + 3\right)\cdot 17^{3} + \left(9 a^{3} + 9 a^{2} + a + 6\right)\cdot 17^{4} + \left(5 a^{3} + a^{2} + 3 a + 2\right)\cdot 17^{5} + \left(15 a^{3} + 7 a^{2} + 11 a + 4\right)\cdot 17^{6} + \left(9 a^{3} + 16 a^{2} + 9 a + 10\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 11 }$ | $=$ | \( 13 a^{3} + 13 a^{2} + 10 a + 11 + \left(14 a^{3} + 3 a^{2} + 14 a + 9\right)\cdot 17 + \left(7 a^{3} + 3 a^{2} + 5 a + 6\right)\cdot 17^{2} + \left(5 a^{3} + 11 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(5 a^{3} + 11 a^{2} + 11 a + 3\right)\cdot 17^{4} + \left(7 a^{3} + 13 a + 16\right)\cdot 17^{5} + \left(7 a^{3} + 12 a^{2} + 16 a + 7\right)\cdot 17^{6} + \left(4 a^{3} + 9 a^{2} + 11 a\right)\cdot 17^{7} +O(17^{8})\) |
$r_{ 12 }$ | $=$ | \( a^{3} + 15 a^{2} + 3 a + 13 + \left(2 a^{3} + 16 a^{2} + 8 a + 2\right)\cdot 17 + \left(3 a^{3} + a^{2} + 16 a + 9\right)\cdot 17^{2} + \left(9 a^{2} + 8 a + 8\right)\cdot 17^{3} + \left(5 a^{3} + 16 a^{2} + 16 a + 9\right)\cdot 17^{4} + \left(11 a^{3} + 15 a^{2} + 9 a + 14\right)\cdot 17^{5} + \left(3 a^{3} + 14 a^{2} + 13 a + 6\right)\cdot 17^{6} + \left(6 a^{3} + 11 a^{2} + 2 a + 4\right)\cdot 17^{7} +O(17^{8})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,8)(3,11)(4,12)(5,9)(7,10)$ | $-2$ |
$3$ | $2$ | $(1,12)(3,9)(4,6)(5,11)$ | $0$ |
$3$ | $2$ | $(1,4)(2,8)(3,5)(6,12)(7,10)(9,11)$ | $0$ |
$2$ | $3$ | $(1,2,12)(3,9,10)(4,6,8)(5,7,11)$ | $-1$ |
$1$ | $4$ | $(1,5,6,9)(2,7,8,10)(3,12,11,4)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,9,6,5)(2,10,8,7)(3,4,11,12)$ | $-2 \zeta_{4}$ |
$3$ | $4$ | $(1,9,6,5)(2,3,8,11)(4,7,12,10)$ | $0$ |
$3$ | $4$ | $(1,5,6,9)(2,11,8,3)(4,10,12,7)$ | $0$ |
$2$ | $6$ | $(1,4,2,6,12,8)(3,7,9,11,10,5)$ | $1$ |
$2$ | $12$ | $(1,10,4,5,2,3,6,7,12,9,8,11)$ | $\zeta_{4}$ |
$2$ | $12$ | $(1,7,4,9,2,11,6,10,12,5,8,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.