Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Artin stem field: | Galois closure of 12.0.110075314176.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.72.6t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.648.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 4x^{11} + 8x^{10} - 12x^{9} + 13x^{8} - 8x^{7} + 4x^{5} - 2x^{4} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{6} + 10x^{3} + 11x^{2} + 11x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a^{5} + 12 a^{4} + a^{2} + 10 a + 5 + \left(5 a^{5} + 4 a^{4} + a^{3} + 10 a^{2} + 3 a + 3\right)\cdot 13 + \left(5 a^{5} + 9 a^{4} + 11 a^{3} + 4 a^{2} + 5 a\right)\cdot 13^{2} + \left(7 a^{5} + 11 a^{4} + 6 a^{3} + a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(7 a^{5} + 4 a^{4} + 11 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 13^{4} + \left(8 a^{5} + 8 a^{4} + 6 a^{3} + 10 a^{2} + 12\right)\cdot 13^{5} + \left(5 a^{5} + 12 a^{3} + 5 a^{2} + 2 a + 3\right)\cdot 13^{6} + \left(2 a^{5} + 2 a^{4} + 5 a^{3} + 12 a^{2} + a + 4\right)\cdot 13^{7} + \left(9 a^{5} + 2 a^{4} + 6 a^{3} + 4 a^{2} + 6\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{5} + 3 a^{4} + 6 a^{3} + a^{2} + 10 a + 1 + \left(10 a^{5} + 9 a^{4} + 9 a^{3} + 6 a^{2} + 5 a + 1\right)\cdot 13 + \left(7 a^{4} + 10 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13^{2} + \left(10 a^{5} + 7 a^{4} + 6 a^{3} + 12 a^{2} + a + 10\right)\cdot 13^{3} + \left(3 a^{5} + 10 a^{4} + 6 a^{3} + 6 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(2 a^{5} + 7 a^{3} + a^{2} + 2 a + 10\right)\cdot 13^{5} + \left(6 a^{5} + 2 a^{4} + 8 a^{3} + 6 a^{2} + a + 5\right)\cdot 13^{6} + \left(a^{5} + 8 a^{4} + 9 a^{3} + 12 a^{2} + 3 a + 2\right)\cdot 13^{7} + \left(2 a^{5} + 10 a^{4} + 4 a^{3} + 6 a^{2} + 12 a + 10\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 3 }$ | $=$ | \( 11 a^{5} + 2 a^{4} + 3 a^{3} + 5 a^{2} + 5 a + 9 + \left(5 a^{5} + 9 a^{4} + 10 a^{3} + 5 a^{2} + 4\right)\cdot 13 + \left(8 a^{5} + 3 a^{4} + 7 a^{3} + 2 a^{2} + 5\right)\cdot 13^{2} + \left(7 a^{5} + 3 a^{4} + 9 a^{3} + 2 a^{2} + 3 a + 1\right)\cdot 13^{3} + \left(11 a^{5} + 2 a^{4} + 6 a^{3} + 6 a^{2} + 6 a + 2\right)\cdot 13^{4} + \left(6 a^{5} + 12 a^{3} + 6 a^{2} + 2 a + 3\right)\cdot 13^{5} + \left(7 a^{5} + 2 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(a^{5} + 10 a^{4} + 7 a^{3} + 3 a^{2} + 5 a + 5\right)\cdot 13^{7} + \left(11 a^{5} + 4 a^{4} + 8 a^{3} + 6 a^{2} + 9 a\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{5} + 12 a^{4} + 12 a^{3} + 2 a^{2} + 2 a + 8 + \left(11 a^{5} + 12 a^{4} + 3 a^{3} + 10 a^{2} + 4 a + 12\right)\cdot 13 + \left(8 a^{5} + 2 a^{4} + 8 a^{3} + a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(3 a^{5} + 12 a^{4} + 11 a^{3} + 12 a^{2} + a + 3\right)\cdot 13^{3} + \left(a^{5} + 4 a^{4} + 6 a^{3} + 4 a^{2} + 12 a + 8\right)\cdot 13^{4} + \left(2 a^{5} + 3 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 1\right)\cdot 13^{5} + \left(9 a^{5} + 7 a^{4} + 5 a^{3} + 4 a^{2} + 11\right)\cdot 13^{6} + \left(2 a^{5} + 2 a^{4} + 8 a^{3} + 9 a^{2} + 7\right)\cdot 13^{7} + \left(2 a^{5} + 10 a^{4} + 11 a^{3} + 3 a^{2} + 9 a + 10\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 5 }$ | $=$ | \( 10 a^{5} + 3 a^{4} + 4 a^{3} + 6 a^{2} + 7 a + 10 + \left(2 a^{5} + 8 a^{4} + 9 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 13 + \left(11 a^{5} + 8 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 13^{2} + \left(2 a^{5} + 7 a^{4} + 8 a^{3} + 3 a^{2} + 4 a + 8\right)\cdot 13^{3} + \left(6 a^{4} + 10 a^{3} + a^{2} + 7\right)\cdot 13^{4} + \left(3 a^{5} + 9 a^{4} + 8 a^{3} + 12 a^{2} + 4 a + 4\right)\cdot 13^{5} + \left(7 a^{4} + 8 a^{3} + a + 7\right)\cdot 13^{6} + \left(7 a^{5} + a^{4} + 3 a^{2} + 5 a + 3\right)\cdot 13^{7} + \left(11 a^{5} + 4 a^{4} + 12 a^{2} + 9 a + 2\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 6 }$ | $=$ | \( 10 a^{5} + 3 a^{4} + 11 a^{3} + a^{2} + a + 7 + \left(6 a^{5} + 9 a^{4} + 11 a^{3} + 12 a + 9\right)\cdot 13 + \left(8 a^{5} + 11 a^{4} + 10 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 13^{2} + \left(4 a^{5} + 7 a^{4} + 6 a^{3} + 10 a^{2} + 10 a\right)\cdot 13^{3} + \left(9 a^{5} + 10 a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(10 a^{5} + 6 a^{4} + 9 a^{3} + 12 a^{2} + 4 a + 7\right)\cdot 13^{5} + \left(2 a^{5} + 11 a^{4} + 5 a^{3} + 9 a^{2} + 11 a + 6\right)\cdot 13^{6} + \left(4 a^{5} + 11 a^{4} + 7 a^{3} + 3 a\right)\cdot 13^{7} + \left(8 a^{5} + 6 a^{4} + 11 a^{2} + 8 a + 6\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 7 }$ | $=$ | \( 3 a^{5} + 5 a^{4} + 5 a^{3} + 3 a^{2} + 3 a + 12 + \left(12 a^{5} + 5 a^{4} + 11 a^{3} + 4 a^{2} + 8 a\right)\cdot 13 + \left(a^{5} + 12 a^{4} + 4 a^{3} + 3 a^{2} + 12 a + 7\right)\cdot 13^{2} + \left(4 a^{5} + a^{4} + 11 a^{2} + 11 a + 6\right)\cdot 13^{3} + \left(8 a^{5} + 7 a^{4} + 8 a^{3} + 6 a^{2} + 10 a + 2\right)\cdot 13^{4} + \left(4 a^{5} + 3 a^{4} + 11 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 13^{5} + \left(9 a^{5} + 6 a^{4} + a^{3} + 4 a^{2} + 2 a + 12\right)\cdot 13^{6} + \left(7 a^{5} + a^{4} + 8 a^{3} + 6 a^{2} + 2 a\right)\cdot 13^{7} + \left(8 a^{5} + 10 a^{4} + a^{3} + 10 a^{2} + 12 a + 9\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 8 }$ | $=$ | \( 2 a^{5} + 11 a^{4} + 6 a^{3} + 9 a^{2} + 4 a + 2 + \left(a^{5} + 8 a^{4} + 11 a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 13 + \left(11 a^{5} + 3 a^{4} + 11 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(4 a^{5} + a^{4} + a^{3} + 4 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(a^{5} + 9 a^{4} + 7 a^{2} + 9 a + 2\right)\cdot 13^{4} + \left(8 a^{5} + 11 a^{4} + 10 a^{3} + 5 a^{2} + 3 a + 11\right)\cdot 13^{5} + \left(7 a^{5} + a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 13^{6} + \left(4 a^{5} + 10 a^{4} + 10 a^{3} + 11 a^{2} + a + 8\right)\cdot 13^{7} + \left(10 a^{4} + 3 a^{3} + 5 a^{2} + 3 a + 5\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 9 }$ | $=$ | \( 12 a^{5} + 12 a^{4} + 3 a^{3} + 3 a^{2} + 3 a + 7 + \left(10 a^{5} + 5 a^{4} + 11 a^{3} + 3 a^{2} + a + 6\right)\cdot 13 + \left(7 a^{5} + 8 a^{4} + 8 a^{3} + 3 a^{2} + 7 a + 7\right)\cdot 13^{2} + \left(6 a^{5} + a^{4} + 3 a^{3} + 4 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(a^{5} + 6 a^{4} + 12 a^{3} + 7 a^{2} + 5 a\right)\cdot 13^{4} + \left(7 a^{5} + 10 a^{4} + 7 a^{3} + 3 a^{2} + 2 a + 8\right)\cdot 13^{5} + \left(11 a^{5} + 9 a^{4} + 4 a^{3} + 8 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(10 a^{5} + a^{4} + 4 a^{2} + 2 a + 1\right)\cdot 13^{7} + \left(3 a^{5} + 8 a^{4} + 2 a^{3} + 4 a^{2} + 3 a + 12\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 10 }$ | $=$ | \( 11 a^{5} + 8 a^{4} + 5 a^{2} + 11 a + 12 + \left(6 a^{4} + 5 a^{3} + 3 a^{2} + 3 a + 2\right)\cdot 13 + \left(a^{5} + 5 a^{4} + 8 a^{3} + 10 a^{2} + 3 a + 10\right)\cdot 13^{2} + \left(5 a^{5} + 6 a^{4} + 12 a^{3} + a^{2} + 8 a + 11\right)\cdot 13^{3} + \left(9 a^{4} + 11 a^{3} + 5 a + 6\right)\cdot 13^{4} + \left(11 a^{5} + 6 a^{4} + 12 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 13^{5} + \left(a^{5} + 3 a^{4} + 12 a^{3} + 8 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(2 a^{5} + 2 a^{4} + 7 a^{3} + 12 a^{2} + 11 a\right)\cdot 13^{7} + \left(11 a^{5} + 9 a^{4} + 8 a^{3} + 10 a^{2} + 12 a + 5\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 11 }$ | $=$ | \( 6 a^{5} + 2 a^{4} + 4 a^{3} + 5 a^{2} + 11 a + 5 + \left(7 a^{5} + 9 a^{4} + 12 a^{3} + 9 a^{2} + 10\right)\cdot 13 + \left(a^{5} + 9 a^{4} + 2 a^{3} + 3 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(7 a^{4} + 6 a^{3} + 2 a^{2} + 3 a + 6\right)\cdot 13^{3} + \left(11 a^{5} + 9 a^{4} + 3 a^{3} + 6 a^{2} + 9 a + 10\right)\cdot 13^{4} + \left(8 a^{5} + 8 a^{4} + 11 a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(8 a^{5} + 8 a^{4} + 11 a^{3} + 9 a^{2} + 5 a + 10\right)\cdot 13^{6} + \left(a^{5} + 7 a^{4} + 5 a^{2} + a + 6\right)\cdot 13^{7} + \left(12 a^{5} + 2 a^{4} + 8 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{8} +O(13^{9})\) |
$r_{ 12 }$ | $=$ | \( 5 a^{5} + 5 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 4 + \left(2 a^{5} + a^{4} + 6 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 13 + \left(11 a^{5} + 7 a^{4} + 3 a^{2} + 7 a\right)\cdot 13^{2} + \left(7 a^{5} + 8 a^{4} + 4 a^{3} + 11 a^{2} + 1\right)\cdot 13^{3} + \left(8 a^{5} + 9 a^{4} + a^{2} + a\right)\cdot 13^{4} + \left(4 a^{5} + 7 a^{4} + 6 a^{3} + a^{2} + 12 a\right)\cdot 13^{5} + \left(7 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(5 a^{5} + 5 a^{4} + 10 a^{3} + 8 a^{2} + a + 8\right)\cdot 13^{7} + \left(10 a^{5} + 11 a^{4} + 8 a^{3} + 12 a^{2} + 3 a + 12\right)\cdot 13^{8} +O(13^{9})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,10)(2,11)(3,6)(4,7)(5,8)(9,12)$ | $-2$ |
$3$ | $2$ | $(1,6)(2,7)(3,10)(4,11)(5,9)(8,12)$ | $0$ |
$3$ | $2$ | $(1,3)(2,4)(5,12)(6,10)(7,11)(8,9)$ | $0$ |
$1$ | $3$ | $(1,12,2)(3,5,4)(6,8,7)(9,11,10)$ | $-2 \zeta_{3} - 2$ |
$1$ | $3$ | $(1,2,12)(3,4,5)(6,7,8)(9,10,11)$ | $2 \zeta_{3}$ |
$2$ | $3$ | $(1,12,2)(3,4,5)(6,7,8)(9,11,10)$ | $-1$ |
$2$ | $3$ | $(3,4,5)(6,7,8)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(3,5,4)(6,8,7)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,9,2,10,12,11)(3,8,4,6,5,7)$ | $2 \zeta_{3} + 2$ |
$1$ | $6$ | $(1,11,12,10,2,9)(3,7,5,6,4,8)$ | $-2 \zeta_{3}$ |
$2$ | $6$ | $(1,11,12,10,2,9)(3,8,4,6,5,7)$ | $1$ |
$2$ | $6$ | $(1,10)(2,11)(3,7,5,6,4,8)(9,12)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,10)(2,11)(3,8,4,6,5,7)(9,12)$ | $\zeta_{3}$ |
$3$ | $6$ | $(1,7,12,6,2,8)(3,11,5,10,4,9)$ | $0$ |
$3$ | $6$ | $(1,8,2,6,12,7)(3,9,4,10,5,11)$ | $0$ |
$3$ | $6$ | $(1,4,12,3,2,5)(6,11,8,10,7,9)$ | $0$ |
$3$ | $6$ | $(1,5,2,3,12,4)(6,9,7,10,8,11)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.