Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(675\)\(\medspace = 3^{3} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 12.0.1037970703125.1 |
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 3.1.135.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 2x^{9} + 4x^{6} - 3x^{3} + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 11 a^{3} + 12 a^{2} + a + 7 + \left(12 a^{3} + a^{2} + 2 a + 8\right)\cdot 13 + \left(5 a^{3} + 10 a^{2} + 11 a + 9\right)\cdot 13^{2} + \left(3 a^{3} + 6 a^{2} + 6 a + 10\right)\cdot 13^{3} + \left(3 a^{3} + 5 a^{2} + 8 a + 11\right)\cdot 13^{4} + \left(6 a^{3} + a^{2} + 6 a + 6\right)\cdot 13^{5} + \left(10 a^{3} + a^{2} + 9 a + 4\right)\cdot 13^{6} + \left(8 a^{3} + 4 a^{2} + 11 a + 1\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 2 }$ | $=$ | \( 4 a^{3} + 9 a^{2} + 5 a + 12 + \left(8 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 13 + \left(9 a^{3} + 11 a^{2} + 10 a + 6\right)\cdot 13^{2} + \left(5 a^{3} + 6 a^{2} + 8 a + 6\right)\cdot 13^{3} + \left(9 a^{3} + 2 a^{2} + 10 a + 2\right)\cdot 13^{4} + \left(6 a^{3} + 5 a^{2} + 11 a + 1\right)\cdot 13^{5} + \left(10 a^{3} + 4 a^{2} + 2 a + 3\right)\cdot 13^{6} + \left(6 a^{3} + 12 a^{2} + 8 a + 5\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 3 }$ | $=$ | \( 7 a^{3} + 6 a^{2} + 3 a + 5 + \left(9 a^{3} + 10 a^{2} + 8 a + 9\right)\cdot 13 + \left(6 a^{3} + 6 a^{2} + 8 a\right)\cdot 13^{2} + \left(11 a^{3} + 10 a^{2} + 5 a + 4\right)\cdot 13^{3} + \left(11 a^{3} + 10 a^{2} + 5 a + 1\right)\cdot 13^{4} + \left(10 a^{3} + 4 a^{2} + 2 a + 3\right)\cdot 13^{5} + \left(7 a^{3} + 12 a^{2} + 2 a + 12\right)\cdot 13^{6} + \left(11 a^{3} + 3 a + 10\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{3} + 5 a^{2} + 4 a + 6 + \left(12 a^{3} + 11 a^{2} + 5 a + 11\right)\cdot 13 + \left(3 a^{3} + 3 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(8 a^{3} + a^{2} + 7 a + 8\right)\cdot 13^{3} + \left(11 a^{3} + 2 a^{2} + 8\right)\cdot 13^{4} + \left(6 a^{3} + 4 a^{2} + 7\right)\cdot 13^{5} + \left(4 a^{3} + 10 a^{2} + 8 a + 3\right)\cdot 13^{6} + \left(a^{3} + 5 a^{2} + 10 a + 11\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 5 }$ | $=$ | \( 7 a^{3} + 10 a^{2} + 3 a + 8 + \left(3 a^{3} + 7 a^{2} + 4 a + 11\right)\cdot 13 + \left(4 a^{3} + 6 a^{2} + 10 a + 8\right)\cdot 13^{2} + \left(5 a^{3} + 4 a^{2} + 8 a + 5\right)\cdot 13^{3} + \left(3 a^{3} + 6 a^{2} + 11 a + 12\right)\cdot 13^{4} + \left(7 a^{3} + 11 a + 11\right)\cdot 13^{5} + \left(2 a^{3} + 2 a^{2} + a + 5\right)\cdot 13^{6} + \left(2 a^{3} + 11 a^{2} + a + 7\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 6 }$ | $=$ | \( 12 a^{3} + a^{2} + 2 a + 10 + \left(3 a^{3} + a^{2} + 3 a + 9\right)\cdot 13 + \left(a^{3} + 12 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(a^{3} + 9 a^{2} + 3 a + 6\right)\cdot 13^{3} + \left(5 a^{3} + a^{2} + 5 a + 4\right)\cdot 13^{4} + \left(12 a^{3} + 11 a^{2} + a + 6\right)\cdot 13^{5} + \left(a^{3} + a^{2} + 6 a + 8\right)\cdot 13^{6} + \left(6 a^{3} + 4 a^{2} + 5 a + 5\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 7 }$ | $=$ | \( 8 a^{3} + 5 a^{2} + 9 a + 2 + \left(a^{3} + 6 a^{2} + 5 a + 5\right)\cdot 13 + \left(11 a^{3} + 2 a^{2} + 4 a + 5\right)\cdot 13^{2} + \left(7 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 13^{3} + \left(12 a^{3} + 4 a + 3\right)\cdot 13^{4} + \left(9 a^{3} + 5 a^{2} + 8 a\right)\cdot 13^{5} + \left(9 a^{3} + 4 a^{2} + 10 a + 3\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 3 a + 12\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{3} + 2 a^{2} + 12 a + 5 + \left(2 a^{3} + 11 a^{2} + 7 a + 9\right)\cdot 13 + \left(2 a^{3} + 10 a^{2} + 7 a + 11\right)\cdot 13^{2} + \left(9 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(2 a^{3} + 12 a^{2} + 9 a + 11\right)\cdot 13^{4} + \left(3 a^{3} + 11 a^{2} + a + 2\right)\cdot 13^{5} + \left(3 a^{3} + a^{2} + 3 a + 7\right)\cdot 13^{6} + \left(6 a^{3} + 10 a^{2} + 4 a + 2\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 9 }$ | $=$ | \( 8 a^{3} + 4 a^{2} + 9 a + 11 + \left(9 a^{3} + 3 a^{2} + 6 a + 5\right)\cdot 13 + \left(2 a^{3} + 9 a^{2} + 4 a + 7\right)\cdot 13^{2} + \left(4 a^{3} + a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(6 a^{3} + a^{2} + 5 a + 1\right)\cdot 13^{4} + \left(12 a^{3} + 11 a^{2} + 7 a + 7\right)\cdot 13^{5} + \left(12 a^{3} + 9 a^{2} + a + 2\right)\cdot 13^{6} + \left(a^{3} + 10 a^{2} + 4\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 10 }$ | $=$ | \( 10 a^{3} + 3 a^{2} + 6 a + 4 + \left(a^{2} + 5 a + 9\right)\cdot 13 + \left(2 a^{3} + 2 a^{2} + 10 a + 7\right)\cdot 13^{2} + \left(6 a^{3} + 9 a^{2} + 12\right)\cdot 13^{3} + \left(11 a^{3} + 8 a^{2} + 10 a + 5\right)\cdot 13^{4} + \left(6 a^{3} + 9 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(6 a^{2} + 3 a + 1\right)\cdot 13^{6} + \left(9 a^{2} + 12 a + 2\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 11 }$ | $=$ | \( 11 a^{3} + 2 a^{2} + a + 6 + \left(a^{3} + 9 a^{2} + 12 a + 11\right)\cdot 13 + \left(8 a^{3} + 3 a^{2} + 12 a + 6\right)\cdot 13^{2} + \left(6 a^{3} + 4\right)\cdot 13^{3} + \left(a^{3} + 2 a^{2} + 3 a + 8\right)\cdot 13^{4} + \left(5 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 13^{5} + \left(8 a^{3} + 9 a^{2} + 10\right)\cdot 13^{6} + \left(2 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 12 }$ | $=$ | \( 10 a^{3} + 6 a^{2} + 10 a + 2 + \left(10 a^{3} + 3 a^{2} + 12 a + 5\right)\cdot 13 + \left(6 a^{3} + 11 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(8 a^{3} + 7 a^{2} + 11 a + 5\right)\cdot 13^{3} + \left(11 a^{3} + 11 a^{2} + 2 a + 5\right)\cdot 13^{4} + \left(2 a^{3} + 9 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(5 a^{3} + a + 2\right)\cdot 13^{6} + \left(5 a^{3} + 10 a^{2} + 11 a + 12\right)\cdot 13^{7} +O(13^{8})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,10)(2,5)(3,8)(4,11)(6,9)(7,12)$ | $-2$ |
$3$ | $2$ | $(1,5)(2,10)(4,12)(7,11)$ | $0$ |
$3$ | $2$ | $(1,2)(3,8)(4,7)(5,10)(6,9)(11,12)$ | $0$ |
$2$ | $3$ | $(1,9,5)(2,10,6)(3,7,11)(4,8,12)$ | $-1$ |
$1$ | $4$ | $(1,4,10,11)(2,7,5,12)(3,9,8,6)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,11,10,4)(2,12,5,7)(3,6,8,9)$ | $-2 \zeta_{4}$ |
$3$ | $4$ | $(1,11,10,4)(2,8,5,3)(6,12,9,7)$ | $0$ |
$3$ | $4$ | $(1,4,10,11)(2,3,5,8)(6,7,9,12)$ | $0$ |
$2$ | $6$ | $(1,2,9,10,5,6)(3,4,7,8,11,12)$ | $1$ |
$2$ | $12$ | $(1,3,2,4,9,7,10,8,5,11,6,12)$ | $\zeta_{4}$ |
$2$ | $12$ | $(1,8,2,11,9,12,10,3,5,4,6,7)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.