Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(8100\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 12.0.21523360500000000.3 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3 \times C_4$ |
Parity: | odd |
Determinant: | 1.4.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.1620.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} + 3 x^{10} - 8 x^{9} + 9 x^{8} + 72 x^{7} + 91 x^{6} + 144 x^{5} - 303 x^{4} - 296 x^{3} + 576 x^{2} + 1536 x + 4096 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{4} + 3x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 11 a^{3} + 3 a^{2} + 9 a + 1 + \left(8 a^{3} + 3 a + 9\right)\cdot 13 + \left(7 a^{3} + 7 a^{2} + 8 a + 11\right)\cdot 13^{2} + \left(3 a^{3} + 2 a^{2} + 2 a + 3\right)\cdot 13^{3} + \left(9 a^{3} + 4 a^{2} + 4 a + 10\right)\cdot 13^{4} + \left(a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 13^{5} + \left(11 a^{3} + 2 a\right)\cdot 13^{6} + \left(11 a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 13^{7} +O(13^{8})\)
$r_{ 2 }$ |
$=$ |
\( 4 a^{3} + 10 a^{2} + 11 a + 7 + \left(a^{3} + 11 a^{2} + 3 a + 10\right)\cdot 13 + \left(3 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 13^{2} + \left(6 a^{3} + 2 a^{2} + 8 a + 2\right)\cdot 13^{3} + \left(8 a^{3} + 10 a^{2} + 3 a + 12\right)\cdot 13^{4} + \left(7 a^{3} + 8 a^{2} + 7 a + 7\right)\cdot 13^{5} + \left(6 a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{6} + \left(12 a^{3} + 6 a^{2} + 6 a + 3\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 3 a^{3} + 4 a^{2} + 3 a + 3 + \left(8 a^{3} + 6 a^{2} + 9 a + 5\right)\cdot 13 + \left(8 a^{3} + 7 a^{2} + 4 a + 1\right)\cdot 13^{2} + \left(8 a^{3} + 5 a^{2} + 11 a + 4\right)\cdot 13^{3} + \left(3 a^{3} + 7 a^{2} + 3 a + 4\right)\cdot 13^{4} + \left(11 a^{3} + a^{2} + 6 a\right)\cdot 13^{5} + \left(6 a^{3} + 2 a^{2} + 7\right)\cdot 13^{6} + \left(12 a^{3} + 4 a + 8\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 8 a^{3} + 5 a^{2} + 6 a + 3 + \left(10 a^{3} + 10 a + 12\right)\cdot 13 + \left(6 a^{3} + 10 a + 12\right)\cdot 13^{2} + \left(12 a^{2} + 9\right)\cdot 13^{3} + \left(2 a^{3} + 6 a + 11\right)\cdot 13^{4} + \left(6 a^{3} + a^{2} + a + 8\right)\cdot 13^{5} + \left(4 a^{3} + 2 a^{2} + a + 7\right)\cdot 13^{6} + \left(7 a^{3} + 6 a^{2} + 7 a + 8\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 5 }$ |
$=$ |
\( 9 a^{3} + a + 12 + \left(10 a^{3} + 8 a^{2} + 3\right)\cdot 13 + \left(5 a^{3} + 10 a + 7\right)\cdot 13^{2} + \left(10 a^{3} + 11 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(12 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(4 a^{3} + 5 a^{2} + 9 a + 3\right)\cdot 13^{5} + \left(11 a^{3} + 10 a^{2} + 5 a + 10\right)\cdot 13^{6} + \left(5 a^{3} + 10 a^{2} + 4 a + 12\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a^{3} + a^{2} + 10 a + 2 + \left(12 a^{3} + 6 a^{2} + 8 a + 11\right)\cdot 13 + \left(8 a^{3} + 11 a^{2} + 5 a + 12\right)\cdot 13^{2} + \left(a^{3} + 11 a^{2} + 10\right)\cdot 13^{3} + \left(3 a^{3} + 7 a^{2} + 10 a + 9\right)\cdot 13^{4} + \left(4 a^{3} + a^{2} + a + 3\right)\cdot 13^{5} + \left(2 a^{3} + 9 a^{2} + a + 11\right)\cdot 13^{6} + \left(6 a^{3} + 9 a^{2} + a\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 7 }$ |
$=$ |
\( 12 a^{3} + 2 a^{2} + 6 a + 3 + \left(3 a^{3} + 5 a^{2} + 3 a + 10\right)\cdot 13 + \left(7 a^{3} + 8 a^{2} + 2 a + 12\right)\cdot 13^{2} + \left(6 a^{3} + 6 a^{2} + 10 a + 7\right)\cdot 13^{3} + \left(10 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(9 a^{3} + 2 a^{2} + 9\right)\cdot 13^{5} + \left(6 a^{3} + 10 a^{2} + 2 a\right)\cdot 13^{6} + \left(8 a^{3} + 8 a^{2} + 9 a + 1\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 8 }$ |
$=$ |
\( 11 a^{3} + 6 a^{2} + 6 a + \left(12 a^{3} + 7 a^{2} + 8 a + 3\right)\cdot 13 + \left(5 a^{3} + 8 a^{2} + 5\right)\cdot 13^{2} + \left(12 a^{3} + a + 11\right)\cdot 13^{3} + \left(4 a^{3} + 2 a^{2} + 5 a + 7\right)\cdot 13^{4} + \left(11 a^{3} + a^{2} + 4 a\right)\cdot 13^{5} + \left(8 a^{2} + 11 a + 1\right)\cdot 13^{6} + \left(8 a^{3} + 9 a^{2} + 7 a + 2\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 9 }$ |
$=$ |
\( 3 a^{3} + 8 a^{2} + 8 + \left(5 a^{3} + 8 a + 9\right)\cdot 13 + \left(8 a^{3} + 7 a^{2} + 2 a + 11\right)\cdot 13^{2} + \left(2 a^{3} + 8 a^{2} + a + 10\right)\cdot 13^{3} + \left(6 a^{3} + 10 a^{2} + 12 a + 11\right)\cdot 13^{4} + \left(10 a^{3} + 9 a^{2} + 4 a + 8\right)\cdot 13^{5} + \left(3 a^{3} + a^{2} + 11 a + 7\right)\cdot 13^{6} + \left(7 a^{3} + 3 a^{2} + 7 a + 3\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 10 }$ |
$=$ |
\( 6 a^{3} + 10 a^{2} + 3 a + \left(6 a^{3} + 4 a^{2} + 9 a\right)\cdot 13 + \left(12 a^{3} + 5 a^{2} + 7 a + 7\right)\cdot 13^{2} + \left(11 a^{3} + 12 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(3 a^{3} + 11 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(6 a^{3} + 3 a + 6\right)\cdot 13^{5} + \left(3 a^{3} + 2 a^{2} + 4 a + 2\right)\cdot 13^{6} + \left(8 a^{3} + 5 a^{2} + 4 a + 10\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 11 }$ |
$=$ |
\( 11 a^{3} + 10 a^{2} + 9 a + 6 + \left(11 a^{3} + 6 a^{2} + 12\right)\cdot 13 + \left(3 a^{3} + 5 a^{2} + 8 a + 3\right)\cdot 13^{2} + \left(7 a^{3} + 9 a^{2} + 3 a + 12\right)\cdot 13^{3} + \left(12 a^{3} + 4 a + 5\right)\cdot 13^{4} + \left(6 a^{3} + 3 a^{2} + a + 4\right)\cdot 13^{5} + \left(5 a^{3} + 9 a^{2} + 4 a + 1\right)\cdot 13^{6} + \left(5 a^{3} + 9 a^{2} + 11 a + 7\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 12 }$ |
$=$ |
\( 6 a^{3} + 6 a^{2} + a + 7 + \left(11 a^{3} + 7 a^{2} + 12 a + 3\right)\cdot 13 + \left(11 a^{3} + 4 a^{2} + 12 a\right)\cdot 13^{2} + \left(5 a^{3} + 7 a^{2} + a + 8\right)\cdot 13^{3} + \left(4 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(10 a^{3} + 9 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(a^{3} + 9 a + 4\right)\cdot 13^{6} + \left(10 a^{3} + 11 a^{2} + 9 a + 3\right)\cdot 13^{7} +O(13^{8})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,9)(3,8)(5,7)(6,11)(10,12)$ | $-2$ |
$3$ | $2$ | $(1,4)(2,3)(5,12)(6,11)(7,10)(8,9)$ | $0$ |
$3$ | $2$ | $(2,8)(3,9)(5,10)(7,12)$ | $0$ |
$2$ | $3$ | $(1,5,10)(2,8,11)(3,6,9)(4,7,12)$ | $-1$ |
$1$ | $4$ | $(1,6,4,11)(2,5,9,7)(3,12,8,10)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,11,4,6)(2,7,9,5)(3,10,8,12)$ | $-2 \zeta_{4}$ |
$3$ | $4$ | $(1,8,4,3)(2,7,9,5)(6,10,11,12)$ | $0$ |
$3$ | $4$ | $(1,3,4,8)(2,5,9,7)(6,12,11,10)$ | $0$ |
$2$ | $6$ | $(1,7,10,4,5,12)(2,3,11,9,8,6)$ | $1$ |
$2$ | $12$ | $(1,8,7,6,10,2,4,3,5,11,12,9)$ | $\zeta_{4}$ |
$2$ | $12$ | $(1,3,7,11,10,9,4,8,5,6,12,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.