Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(27075\)\(\medspace = 3 \cdot 5^{2} \cdot 19^{2} \) |
Artin stem field: | Galois closure of 12.4.24181674720486328125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3 \times C_4$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.5415.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 3 x^{11} + 18 x^{10} - 40 x^{9} + 189 x^{8} - 336 x^{7} + 734 x^{6} - 1092 x^{5} + 501 x^{4} + \cdots + 10411 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{4} + 3x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 9 a^{3} + 8 a^{2} + a + 1 + \left(10 a^{3} + 10 a^{2} + 5 a + 9\right)\cdot 13 + \left(9 a^{3} + a^{2} + 5\right)\cdot 13^{2} + \left(2 a^{3} + 4 a^{2} + 8 a + 9\right)\cdot 13^{3} + \left(11 a^{3} + 6 a^{2} + 9 a + 3\right)\cdot 13^{4} + \left(2 a^{3} + 2 a^{2} + 7 a + 9\right)\cdot 13^{5} + \left(4 a^{3} + 7 a^{2} + 2 a + 3\right)\cdot 13^{6} + \left(3 a^{3} + 6 a^{2} + 6 a + 11\right)\cdot 13^{7} + \left(12 a^{3} + 7 a^{2} + a + 8\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{3} + 8 a^{2} + a + 10 + \left(10 a^{3} + 10 a^{2} + 5 a\right)\cdot 13 + \left(9 a^{3} + a^{2} + 1\right)\cdot 13^{2} + \left(2 a^{3} + 4 a^{2} + 8 a + 7\right)\cdot 13^{3} + \left(11 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 13^{4} + \left(2 a^{3} + 2 a^{2} + 7 a + 4\right)\cdot 13^{5} + \left(4 a^{3} + 7 a^{2} + 2 a + 9\right)\cdot 13^{6} + \left(3 a^{3} + 6 a^{2} + 6 a + 9\right)\cdot 13^{7} + \left(12 a^{3} + 7 a^{2} + a + 6\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 3 }$ | $=$ | \( 10 a^{3} + 5 a^{2} + 10 a + 10 + \left(6 a^{3} + 8 a^{2} + 5 a + 11\right)\cdot 13 + \left(9 a^{3} + 9 a^{2} + 11 a + 5\right)\cdot 13^{2} + \left(9 a^{3} + 5 a^{2} + 4 a + 8\right)\cdot 13^{3} + \left(6 a^{3} + 6 a^{2} + 10 a + 8\right)\cdot 13^{4} + \left(10 a^{3} + 5 a^{2} + 7 a + 8\right)\cdot 13^{5} + \left(a^{3} + 4 a^{2} + 12 a + 6\right)\cdot 13^{6} + \left(4 a^{3} + a^{2} + a + 3\right)\cdot 13^{7} + \left(3 a^{3} + 8 a^{2} + 7 a + 6\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 4 }$ | $=$ | \( 8 a^{3} + 2 a^{2} + 11 a + 7 + \left(9 a^{3} + 5 a^{2} + 7 a + 6\right)\cdot 13 + \left(9 a^{3} + 5 a^{2} + 11 a + 1\right)\cdot 13^{2} + \left(7 a^{3} + 6 a^{2} + a + 11\right)\cdot 13^{3} + \left(2 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{4} + \left(4 a^{3} + 3 a^{2} + 12 a + 1\right)\cdot 13^{5} + \left(5 a^{3} + 5 a^{2} + 7 a + 7\right)\cdot 13^{6} + \left(2 a^{3} + 12 a^{2} + 7 a + 4\right)\cdot 13^{7} + \left(8 a^{3} + 11 a^{2} + 8 a + 10\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 5 }$ | $=$ | \( 10 a^{3} + a + 9 + \left(9 a^{3} + 5 a + 6\right)\cdot 13 + \left(4 a^{3} + 4 a^{2} + 3 a + 6\right)\cdot 13^{2} + \left(9 a^{3} + 2 a^{2} + 4 a + 6\right)\cdot 13^{3} + \left(6 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 13^{4} + \left(5 a^{3} + 8 a^{2} + 2 a\right)\cdot 13^{5} + \left(9 a^{3} + 9 a^{2} + 7 a + 5\right)\cdot 13^{6} + \left(12 a^{3} + 3 a^{2} + 2 a + 6\right)\cdot 13^{7} + \left(a^{3} + 7 a^{2} + 5 a + 12\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 6 }$ | $=$ | \( 12 a^{3} + 3 a^{2} + 10 + \left(8 a^{3} + 10 a^{2} + 8 a + 3\right)\cdot 13 + \left(a^{3} + a^{2} + 10 a + 12\right)\cdot 13^{2} + \left(6 a^{3} + 11 a + 11\right)\cdot 13^{3} + \left(5 a^{3} + 3 a^{2} + 10\right)\cdot 13^{4} + \left(11 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(7 a^{3} + 3 a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(7 a^{3} + 3 a^{2} + 9 a + 3\right)\cdot 13^{7} + \left(3 a^{3} + 12 a^{2} + 10 a + 7\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 7 }$ | $=$ | \( 9 a^{3} + 3 a^{2} + 5 a + 9 + \left(2 a^{2} + 4 a + 8\right)\cdot 13 + \left(5 a^{3} + 9 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(2 a^{3} + 7 a + 5\right)\cdot 13^{3} + \left(11 a^{3} + 5 a^{2} + 2 a\right)\cdot 13^{4} + \left(10 a^{3} + 5 a^{2} + 2 a + 9\right)\cdot 13^{5} + \left(11 a^{3} + 12 a^{2} + 10 a + 7\right)\cdot 13^{6} + \left(11 a^{3} + 8 a^{2} + 6 a + 12\right)\cdot 13^{7} + \left(10 a^{3} + 6 a^{2} + 12 a + 12\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 8 }$ | $=$ | \( 9 a^{3} + 12 a^{2} + 7 a + 7 + \left(5 a^{3} + 2 a^{2} + 8 a + 4\right)\cdot 13 + \left(9 a^{3} + 9 a + 6\right)\cdot 13^{2} + \left(a^{3} + 8 a^{2} + 11 a + 12\right)\cdot 13^{3} + \left(11 a^{3} + 11 a^{2} + 12 a + 10\right)\cdot 13^{4} + \left(11 a^{3} + 6 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(2 a^{3} + 2 a^{2} + 4 a + 4\right)\cdot 13^{6} + \left(3 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13^{7} + \left(12 a^{3} + 12 a^{2} + a + 9\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 9 }$ | $=$ | \( 11 a^{3} + 6 a^{2} + 4 a + 1 + \left(12 a^{3} + 12 a^{2} + 7 a + 1\right)\cdot 13 + \left(a^{3} + 6 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(12 a^{3} + 11 a^{2} + a + 12\right)\cdot 13^{3} + \left(9 a^{3} + 2 a^{2} + 5\right)\cdot 13^{4} + \left(5 a^{3} + 8 a^{2} + 3 a + 2\right)\cdot 13^{5} + \left(9 a^{3} + 6 a^{2} + 11 a + 7\right)\cdot 13^{6} + \left(6 a^{3} + 8 a^{2} + 11\right)\cdot 13^{7} + \left(12 a^{3} + 11 a^{2} + 5 a + 9\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 10 }$ | $=$ | \( 12 a^{3} + 3 a^{2} + 1 + \left(8 a^{3} + 10 a^{2} + 8 a + 12\right)\cdot 13 + \left(a^{3} + a^{2} + 10 a + 3\right)\cdot 13^{2} + \left(6 a^{3} + 11 a + 1\right)\cdot 13^{3} + \left(5 a^{3} + 3 a^{2} + 12\right)\cdot 13^{4} + \left(11 a^{2} + 3 a + 6\right)\cdot 13^{5} + \left(7 a^{3} + 3 a^{2} + 8 a + 4\right)\cdot 13^{6} + \left(7 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 13^{7} + \left(3 a^{3} + 12 a^{2} + 10 a + 9\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 11 }$ | $=$ | \( 9 a^{3} + 12 a^{2} + 7 a + 3 + \left(5 a^{3} + 2 a^{2} + 8 a + 9\right)\cdot 13 + \left(9 a^{3} + 9 a + 1\right)\cdot 13^{2} + \left(a^{3} + 8 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(11 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(11 a^{3} + 6 a^{2} + 12 a\right)\cdot 13^{5} + \left(2 a^{3} + 2 a^{2} + 4 a + 10\right)\cdot 13^{6} + \left(3 a^{3} + 7 a^{2} + 3 a + 9\right)\cdot 13^{7} + \left(12 a^{3} + 12 a^{2} + a + 7\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 12 }$ | $=$ | \( 9 a^{3} + 3 a^{2} + 5 a + \left(2 a^{2} + 4 a + 4\right)\cdot 13 + \left(5 a^{3} + 9 a^{2} + 5 a + 6\right)\cdot 13^{2} + \left(2 a^{3} + 7 a + 7\right)\cdot 13^{3} + \left(11 a^{3} + 5 a^{2} + 2 a + 1\right)\cdot 13^{4} + \left(10 a^{3} + 5 a^{2} + 2 a + 1\right)\cdot 13^{5} + \left(11 a^{3} + 12 a^{2} + 10 a + 2\right)\cdot 13^{6} + \left(11 a^{3} + 8 a^{2} + 6 a + 1\right)\cdot 13^{7} + \left(10 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 13^{8} +O(13^{9})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,12)(2,7)(3,5)(4,9)(6,11)(8,10)$ | $-2$ |
$3$ | $2$ | $(1,7)(2,12)(3,5)(4,9)(6,8)(10,11)$ | $0$ |
$3$ | $2$ | $(2,3)(4,8)(5,7)(9,10)$ | $0$ |
$2$ | $3$ | $(1,2,3)(4,11,8)(5,12,7)(6,10,9)$ | $-1$ |
$1$ | $4$ | $(1,11,12,6)(2,8,7,10)(3,4,5,9)$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,6,12,11)(2,10,7,8)(3,9,5,4)$ | $2 \zeta_{4}$ |
$3$ | $4$ | $(1,10,12,8)(2,6,7,11)(3,9,5,4)$ | $0$ |
$3$ | $4$ | $(1,8,12,10)(2,11,7,6)(3,4,5,9)$ | $0$ |
$2$ | $6$ | $(1,7,3,12,2,5)(4,6,8,9,11,10)$ | $1$ |
$2$ | $12$ | $(1,10,5,11,2,9,12,8,3,6,7,4)$ | $-\zeta_{4}$ |
$2$ | $12$ | $(1,8,5,6,2,4,12,10,3,11,7,9)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.