Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(3328\)\(\medspace = 2^{8} \cdot 13 \) |
Artin stem field: | Galois closure of 12.0.981348487528448.5 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3 \times C_4$ |
Parity: | odd |
Determinant: | 1.52.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.104.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} + 4x^{10} + 6x^{8} - 8x^{6} - 7x^{4} + 4x^{2} + 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{4} + 2x^{2} + 11x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 a^{3} + 14 a^{2} + 8 a + 9 + \left(3 a^{3} + 10 a^{2} + 13 a + 13\right)\cdot 19 + \left(12 a^{3} + 2 a^{2} + 16 a + 12\right)\cdot 19^{2} + \left(8 a^{3} + 10 a^{2} + 6 a\right)\cdot 19^{3} + \left(18 a^{3} + 16 a^{2} + 2 a + 12\right)\cdot 19^{4} + \left(8 a^{3} + 14 a^{2} + 17 a + 17\right)\cdot 19^{5} + \left(3 a^{3} + 6 a^{2} + 3 a + 6\right)\cdot 19^{6} + \left(12 a^{3} + 15 a^{2} + a + 11\right)\cdot 19^{7} + \left(16 a^{3} + 12 a^{2} + 7\right)\cdot 19^{8} +O(19^{9})\)
$r_{ 2 }$ |
$=$ |
\( 17 a^{3} + 18 a^{2} + 6 a + 11 + \left(7 a^{3} + 15 a + 18\right)\cdot 19 + \left(13 a^{2} + 9 a + 6\right)\cdot 19^{2} + \left(12 a^{3} + 4 a^{2} + 5 a + 18\right)\cdot 19^{3} + \left(5 a^{3} + 10 a^{2} + 7 a + 13\right)\cdot 19^{4} + \left(2 a + 7\right)\cdot 19^{5} + \left(11 a^{3} + 13 a^{2} + 10 a + 18\right)\cdot 19^{6} + \left(9 a^{3} + 11 a^{2} + 15 a + 9\right)\cdot 19^{7} + \left(17 a^{3} + 11 a^{2} + a + 13\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a^{3} + 17 a^{2} + 16 a + 1 + \left(5 a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 19 + \left(15 a^{3} + 6 a^{2} + 10 a + 14\right)\cdot 19^{2} + \left(13 a^{3} + 13 a^{2} + 8 a + 3\right)\cdot 19^{3} + \left(6 a^{3} + 17 a^{2} + 8 a + 16\right)\cdot 19^{4} + \left(2 a^{3} + 6 a^{2} + 15 a + 16\right)\cdot 19^{5} + \left(13 a^{3} + 14 a^{2} + 18 a + 3\right)\cdot 19^{6} + \left(2 a^{3} + 17 a^{2} + 5 a + 16\right)\cdot 19^{7} + \left(17 a^{3} + 8 a^{2} + 12 a + 7\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 17 a^{3} + a^{2} + 3 a + 13 + \left(12 a^{3} + 9 a^{2} + 7 a + 15\right)\cdot 19 + \left(18 a^{3} + 6 a^{2} + 6 a + 3\right)\cdot 19^{2} + \left(14 a^{3} + 16 a^{2} + 9 a + 2\right)\cdot 19^{3} + \left(5 a^{3} + 6 a^{2} + 3 a + 7\right)\cdot 19^{4} + \left(15 a^{3} + 7 a^{2} + 12 a + 5\right)\cdot 19^{5} + \left(16 a^{3} + 6 a^{2} + 14 a + 7\right)\cdot 19^{6} + \left(13 a^{2} + a + 6\right)\cdot 19^{7} + \left(9 a^{3} + 13 a + 18\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a^{3} + a^{2} + 4 a + 3 + \left(9 a^{3} + 12 a^{2} + 13 a + 8\right)\cdot 19 + \left(15 a^{3} + 10 a^{2} + 17 a + 5\right)\cdot 19^{2} + \left(16 a^{3} + 13 a^{2} + 10 a\right)\cdot 19^{3} + \left(13 a^{3} + 3 a + 6\right)\cdot 19^{4} + \left(17 a^{3} + 8 a^{2} + 7 a + 2\right)\cdot 19^{5} + \left(12 a^{3} + 4 a^{2} + 5 a + 16\right)\cdot 19^{6} + \left(10 a^{3} + 16 a^{2} + 18 a + 18\right)\cdot 19^{7} + \left(6 a^{3} + 6 a^{2} + 8 a + 3\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 2 a^{3} + 16 a^{2} + 4 + \left(9 a^{3} + a^{2} + 2 a + 15\right)\cdot 19 + \left(13 a^{3} + 15 a^{2} + 6 a + 2\right)\cdot 19^{2} + \left(8 a^{3} + 12 a^{2} + 15 a + 18\right)\cdot 19^{3} + \left(10 a^{3} + 16 a^{2} + 9 a + 7\right)\cdot 19^{4} + \left(a^{2} + 16 a + 6\right)\cdot 19^{5} + \left(3 a^{3} + 17 a^{2} + a + 18\right)\cdot 19^{6} + \left(18 a^{3} + 4 a^{2} + 9 a + 16\right)\cdot 19^{7} + \left(2 a^{3} + 9 a^{2} + 3 a + 9\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 7 }$ |
$=$ |
\( 15 a^{3} + 5 a^{2} + 11 a + 10 + \left(15 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 19 + \left(6 a^{3} + 16 a^{2} + 2 a + 6\right)\cdot 19^{2} + \left(10 a^{3} + 8 a^{2} + 12 a + 18\right)\cdot 19^{3} + \left(2 a^{2} + 16 a + 6\right)\cdot 19^{4} + \left(10 a^{3} + 4 a^{2} + a + 1\right)\cdot 19^{5} + \left(15 a^{3} + 12 a^{2} + 15 a + 12\right)\cdot 19^{6} + \left(6 a^{3} + 3 a^{2} + 17 a + 7\right)\cdot 19^{7} + \left(2 a^{3} + 6 a^{2} + 18 a + 11\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 8 }$ |
$=$ |
\( 2 a^{3} + a^{2} + 13 a + 8 + \left(11 a^{3} + 18 a^{2} + 3 a\right)\cdot 19 + \left(18 a^{3} + 5 a^{2} + 9 a + 12\right)\cdot 19^{2} + \left(6 a^{3} + 14 a^{2} + 13 a\right)\cdot 19^{3} + \left(13 a^{3} + 8 a^{2} + 11 a + 5\right)\cdot 19^{4} + \left(18 a^{3} + 18 a^{2} + 16 a + 11\right)\cdot 19^{5} + \left(7 a^{3} + 5 a^{2} + 8 a\right)\cdot 19^{6} + \left(9 a^{3} + 7 a^{2} + 3 a + 9\right)\cdot 19^{7} + \left(a^{3} + 7 a^{2} + 17 a + 5\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 9 }$ |
$=$ |
\( 10 a^{3} + 2 a^{2} + 3 a + 18 + \left(13 a^{3} + 6 a^{2} + 17\right)\cdot 19 + \left(3 a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 19^{2} + \left(5 a^{3} + 5 a^{2} + 10 a + 15\right)\cdot 19^{3} + \left(12 a^{3} + a^{2} + 10 a + 2\right)\cdot 19^{4} + \left(16 a^{3} + 12 a^{2} + 3 a + 2\right)\cdot 19^{5} + \left(5 a^{3} + 4 a^{2} + 15\right)\cdot 19^{6} + \left(16 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{7} + \left(a^{3} + 10 a^{2} + 6 a + 11\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 10 }$ |
$=$ |
\( 2 a^{3} + 18 a^{2} + 16 a + 6 + \left(6 a^{3} + 9 a^{2} + 11 a + 3\right)\cdot 19 + \left(12 a^{2} + 12 a + 15\right)\cdot 19^{2} + \left(4 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(13 a^{3} + 12 a^{2} + 15 a + 11\right)\cdot 19^{4} + \left(3 a^{3} + 11 a^{2} + 6 a + 13\right)\cdot 19^{5} + \left(2 a^{3} + 12 a^{2} + 4 a + 11\right)\cdot 19^{6} + \left(18 a^{3} + 5 a^{2} + 17 a + 12\right)\cdot 19^{7} + \left(9 a^{3} + 18 a^{2} + 5 a\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 11 }$ |
$=$ |
\( 13 a^{3} + 18 a^{2} + 15 a + 16 + \left(9 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 19 + \left(3 a^{3} + 8 a^{2} + a + 13\right)\cdot 19^{2} + \left(2 a^{3} + 5 a^{2} + 8 a + 18\right)\cdot 19^{3} + \left(5 a^{3} + 18 a^{2} + 15 a + 12\right)\cdot 19^{4} + \left(a^{3} + 10 a^{2} + 11 a + 16\right)\cdot 19^{5} + \left(6 a^{3} + 14 a^{2} + 13 a + 2\right)\cdot 19^{6} + \left(8 a^{3} + 2 a^{2}\right)\cdot 19^{7} + \left(12 a^{3} + 12 a^{2} + 10 a + 15\right)\cdot 19^{8} +O(19^{9})\)
| $r_{ 12 }$ |
$=$ |
\( 17 a^{3} + 3 a^{2} + 15 + \left(9 a^{3} + 17 a^{2} + 17 a + 3\right)\cdot 19 + \left(5 a^{3} + 3 a^{2} + 12 a + 16\right)\cdot 19^{2} + \left(10 a^{3} + 6 a^{2} + 3 a\right)\cdot 19^{3} + \left(8 a^{3} + 2 a^{2} + 9 a + 11\right)\cdot 19^{4} + \left(18 a^{3} + 17 a^{2} + 2 a + 12\right)\cdot 19^{5} + \left(15 a^{3} + a^{2} + 17 a\right)\cdot 19^{6} + \left(14 a^{2} + 9 a + 2\right)\cdot 19^{7} + \left(16 a^{3} + 9 a^{2} + 15 a + 9\right)\cdot 19^{8} +O(19^{9})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-2$ |
$3$ | $2$ | $(2,5)(3,12)(6,9)(8,11)$ | $0$ |
$3$ | $2$ | $(1,7)(2,11)(3,6)(4,10)(5,8)(9,12)$ | $0$ |
$2$ | $3$ | $(1,11,8)(2,7,5)(3,10,12)(4,6,9)$ | $-1$ |
$1$ | $4$ | $(1,10,7,4)(2,9,8,3)(5,6,11,12)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,4,7,10)(2,3,8,9)(5,12,11,6)$ | $-2 \zeta_{4}$ |
$3$ | $4$ | $(1,10,7,4)(2,6,8,12)(3,5,9,11)$ | $0$ |
$3$ | $4$ | $(1,4,7,10)(2,12,8,6)(3,11,9,5)$ | $0$ |
$2$ | $6$ | $(1,2,11,7,8,5)(3,6,10,9,12,4)$ | $1$ |
$2$ | $12$ | $(1,6,2,10,11,9,7,12,8,4,5,3)$ | $\zeta_{4}$ |
$2$ | $12$ | $(1,12,2,4,11,3,7,6,8,10,5,9)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.