Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
Artin stem field: | Galois closure of 12.0.9834496000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.280.6t1.d.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.1960.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 2x^{11} - x^{10} - 2x^{9} + 9x^{8} + 2x^{7} - 4x^{6} - 8x^{5} + 2x^{4} + 8x^{3} + 7x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{6} + 10x^{3} + 11x^{2} + 11x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 5 a^{5} + 5 a^{4} + 2 a^{3} + 2 a^{2} + 4 a + 6 + \left(5 a^{5} + a^{4} + 6 a^{3} + 10 a^{2} + 10 a + 1\right)\cdot 13 + \left(4 a^{5} + 7 a^{4} + a^{3} + 8 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(3 a^{5} + 10 a^{4} + 11 a^{3} + a^{2} + 11\right)\cdot 13^{3} + \left(a^{5} + 5 a^{4} + 4 a^{3} + 4 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(6 a^{5} + 5 a^{4} + 6 a^{3} + 5 a^{2} + 7 a + 8\right)\cdot 13^{5} + \left(2 a^{5} + 2 a^{4} + 7 a^{3} + 4 a^{2} + 9 a + 10\right)\cdot 13^{6} +O(13^{7})\)
$r_{ 2 }$ |
$=$ |
\( 3 a^{5} + 10 a^{4} + 6 a^{3} + 9 a^{2} + a + 1 + \left(8 a^{5} + 9 a^{4} + 11 a^{3} + 2 a^{2} + 3 a + 5\right)\cdot 13 + \left(6 a^{5} + 9 a^{4} + 9 a^{3} + 10 a^{2} + 4 a + 8\right)\cdot 13^{2} + \left(12 a^{5} + 2 a^{4} + 4 a^{3} + 5 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(7 a^{5} + 5 a^{4} + 10 a^{3} + 3 a^{2} + 3 a + 9\right)\cdot 13^{4} + \left(3 a^{5} + 10 a^{4} + 7 a^{3} + 12 a^{2} + 9\right)\cdot 13^{5} + \left(7 a^{5} + 6 a^{4} + 9 a^{3} + 9 a^{2} + 11 a + 10\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 10 a^{4} + 12 a^{3} + 8 a^{2} + 10 a + 10 + \left(5 a^{5} + 10 a^{4} + 7 a^{3} + 11 a^{2} + 5 a + 12\right)\cdot 13 + \left(8 a^{5} + 2 a^{4} + 2 a^{3} + 10 a^{2} + 6 a + 10\right)\cdot 13^{2} + \left(10 a^{5} + 8 a^{4} + 10 a^{3} + 3 a^{2} + 5 a\right)\cdot 13^{3} + \left(2 a^{5} + 3 a^{4} + 9 a^{3} + 9 a^{2} + 4 a + 6\right)\cdot 13^{4} + \left(9 a^{5} + 2 a^{3} + 2 a^{2} + 4 a + 11\right)\cdot 13^{5} + \left(2 a^{5} + 9 a^{4} + 8 a^{3} + 8 a^{2} + 12 a + 12\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 4 a^{5} + 9 a^{4} + 6 a^{3} + 12 a^{2} + 11 a + 5 + \left(4 a^{5} + 11 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 13 + \left(7 a^{5} + 7 a^{4} + 5 a^{3} + 10 a^{2} + a + 5\right)\cdot 13^{2} + \left(7 a^{5} + 4 a^{4} + 7 a^{3} + 2 a^{2} + 4 a + 3\right)\cdot 13^{3} + \left(9 a^{4} + 11 a^{3} + 6 a^{2} + 6 a + 8\right)\cdot 13^{4} + \left(a^{5} + 10 a^{4} + 3 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(2 a^{5} + 5 a^{4} + 11 a^{3} + 9 a^{2} + 6 a\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a^{5} + 5 a^{4} + a^{3} + 12 a^{2} + 6 a + 8 + \left(5 a^{5} + 6 a^{4} + a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 13 + \left(6 a^{4} + 3 a^{3} + 11 a^{2} + 6\right)\cdot 13^{2} + \left(8 a^{5} + 8 a^{4} + 2 a^{3} + 11 a^{2} + 3 a + 8\right)\cdot 13^{3} + \left(4 a^{5} + 11 a^{4} + 7 a^{2} + 6 a + 11\right)\cdot 13^{4} + \left(a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + a + 9\right)\cdot 13^{5} + \left(12 a^{5} + 7 a^{4} + 11 a^{3} + 8 a^{2} + 11 a + 9\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 4 a^{5} + a^{4} + 7 a^{3} + a^{2} + 12 a + 12 + \left(9 a^{5} + 3 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 3\right)\cdot 13 + \left(6 a^{5} + 6 a^{4} + 10 a^{3} + 7 a^{2} + 8 a + 9\right)\cdot 13^{2} + \left(6 a^{5} + a^{4} + 5 a^{3} + a^{2} + 10 a + 8\right)\cdot 13^{3} + \left(3 a^{5} + 8 a^{4} + 4 a^{3} + 12 a^{2} + 7\right)\cdot 13^{4} + \left(5 a^{5} + 9 a^{4} + 9 a^{3} + a + 9\right)\cdot 13^{5} + \left(10 a^{5} + 3 a^{4} + 10 a^{3} + 11 a^{2} + 6 a + 11\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 7 }$ |
$=$ |
\( 12 a^{5} + 4 a^{4} + 4 a^{3} + 2 a^{2} + 7 a + 10 + \left(2 a^{4} + 5 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 13 + \left(6 a^{5} + 2 a^{4} + 3 a^{3} + 6 a^{2} + 8 a + 12\right)\cdot 13^{2} + \left(10 a^{5} + 9 a^{4} + 6 a^{3} + 2 a^{2} + 7 a + 5\right)\cdot 13^{3} + \left(4 a^{4} + 10 a^{3} + 4 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(11 a^{5} + 9 a^{4} + a^{3} + 4 a + 1\right)\cdot 13^{5} + \left(3 a^{5} + 5 a^{4} + 5 a^{3} + 11 a^{2} + 9 a + 1\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 8 }$ |
$=$ |
\( 11 a^{5} + 8 a^{4} + 3 a^{3} + 11 a^{2} + 5 a + 10 + \left(8 a^{5} + 12 a^{4} + 10 a^{3} + 2 a^{2} + 4 a + 3\right)\cdot 13 + \left(8 a^{5} + 4 a^{4} + 8 a^{3} + 2 a^{2} + 7 a + 8\right)\cdot 13^{2} + \left(11 a^{5} + 11 a^{4} + 3 a^{3} + 8 a^{2} + 3 a + 11\right)\cdot 13^{3} + \left(2 a^{5} + 8 a^{3} + 11 a^{2} + 4 a + 2\right)\cdot 13^{4} + \left(11 a^{5} + 8 a^{4} + 12 a^{3} + 8 a + 4\right)\cdot 13^{5} + \left(9 a^{5} + 3 a^{4} + 9 a^{3} + 3 a^{2} + 3 a + 4\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 9 }$ |
$=$ |
\( 6 a^{5} + 2 a^{4} + 3 a^{2} + 7 + \left(12 a^{5} + 12 a^{4} + 10 a^{3} + a^{2} + 8\right)\cdot 13 + \left(7 a^{5} + 12 a^{4} + 8 a^{3} + 12 a^{2} + 12 a + 3\right)\cdot 13^{2} + \left(3 a^{5} + 9 a^{4} + 7 a^{3} + 6 a^{2} + 7 a + 8\right)\cdot 13^{3} + \left(7 a^{5} + 3 a^{4} + 8 a^{2} + 8 a + 10\right)\cdot 13^{4} + \left(5 a^{5} + 8 a^{4} + 10 a^{3} + 3 a^{2} + 8 a + 3\right)\cdot 13^{5} + \left(7 a^{5} + 5 a^{3} + 11 a^{2} + 11 a + 6\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 10 }$ |
$=$ |
\( 8 a^{5} + 5 a^{4} + 12 a^{3} + 9 a + 12 + \left(3 a^{5} + 11 a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 3\right)\cdot 13 + \left(11 a^{5} + 12 a^{4} + 8 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 13^{2} + \left(9 a^{5} + 11 a^{4} + 12 a^{3} + 3 a^{2} + 5 a + 3\right)\cdot 13^{3} + \left(11 a^{4} + 7 a^{3} + 6 a^{2} + 4 a + 4\right)\cdot 13^{4} + \left(3 a^{5} + 12 a^{4} + 8 a^{3} + 2 a^{2} + 8 a + 3\right)\cdot 13^{5} + \left(a^{5} + 10 a^{4} + 8 a^{3} + 12 a^{2} + 8 a + 5\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 11 }$ |
$=$ |
\( 4 a^{5} + 11 a^{4} + a^{3} + 11 a^{2} + 2 a + 12 + \left(6 a^{5} + 3 a^{4} + 8 a^{3} + 6 a^{2} + 2 a + 2\right)\cdot 13 + \left(9 a^{5} + 3 a^{4} + 3 a^{3} + 3 a^{2} + 7 a + 12\right)\cdot 13^{2} + \left(12 a^{5} + 7 a^{4} + 12 a^{3} + 3 a^{2} + 5 a + 5\right)\cdot 13^{3} + \left(4 a^{5} + 8 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(3 a^{5} + 6 a^{4} + 8 a^{2} + 1\right)\cdot 13^{5} + \left(4 a^{5} + 6 a^{4} + 8 a^{3} + 6 a^{2} + 7 a + 12\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 12 }$ |
$=$ |
\( 2 a^{5} + 8 a^{4} + 11 a^{3} + 7 a^{2} + 11 a + \left(8 a^{5} + 3 a^{4} + 2 a^{3} + 11 a^{2} + 12 a + 5\right)\cdot 13 + \left(2 a^{4} + 11 a^{3} + 8 a^{2} + 12 a + 11\right)\cdot 13^{2} + \left(7 a^{5} + 5 a^{4} + 6 a^{3} + 12 a^{2} + a + 6\right)\cdot 13^{3} + \left(a^{5} + 4 a^{4} + 5 a^{3} + 6 a^{2} + 4 a + 2\right)\cdot 13^{4} + \left(4 a^{5} + 2 a^{4} + 9 a^{3} + 5 a^{2} + 3 a\right)\cdot 13^{5} + \left(a^{5} + 2 a^{4} + 7 a^{3} + 8 a^{2} + 7 a + 6\right)\cdot 13^{6} +O(13^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-2$ |
$3$ | $2$ | $(1,12)(2,10)(3,5)(4,8)(6,7)(9,11)$ | $0$ |
$3$ | $2$ | $(1,2)(3,10)(4,9)(5,6)(7,8)(11,12)$ | $0$ |
$1$ | $3$ | $(1,5,4)(2,6,9)(3,8,12)(7,11,10)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,4,5)(2,9,6)(3,12,8)(7,10,11)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(1,4,5)(7,10,11)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,5,4)(7,11,10)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,5,4)(2,9,6)(3,12,8)(7,11,10)$ | $-1$ |
$1$ | $6$ | $(1,11,4,7,5,10)(2,12,9,8,6,3)$ | $-2 \zeta_{3}$ |
$1$ | $6$ | $(1,10,5,7,4,11)(2,3,6,8,9,12)$ | $2 \zeta_{3} + 2$ |
$2$ | $6$ | $(1,11,4,7,5,10)(2,8)(3,9)(6,12)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,10,5,7,4,11)(2,8)(3,9)(6,12)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,10,5,7,4,11)(2,12,9,8,6,3)$ | $1$ |
$3$ | $6$ | $(1,8,5,12,4,3)(2,11,6,10,9,7)$ | $0$ |
$3$ | $6$ | $(1,3,4,12,5,8)(2,7,9,10,6,11)$ | $0$ |
$3$ | $6$ | $(1,6,4,2,5,9)(3,7,12,10,8,11)$ | $0$ |
$3$ | $6$ | $(1,9,5,2,4,6)(3,11,8,10,12,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.