Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(2205\)\(\medspace = 3^{2} \cdot 5 \cdot 7^{2} \) |
Artin stem field: | Galois closure of 12.0.28958126698265625.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.315.6t1.c.b |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.2835.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} + 5x^{10} - 7x^{9} + 46x^{8} + 49x^{7} + 258x^{6} + 154x^{5} + 310x^{4} + 35x^{3} + 115x^{2} + 14x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{6} + 17x^{3} + 17x^{2} + 6x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{5} + 15 a^{4} + 7 a^{3} + 10 a^{2} + 6 a + 11 + \left(17 a^{5} + 14 a^{4} + 10 a^{3} + 16 a^{2} + 18 a + 18\right)\cdot 19 + \left(14 a^{5} + 15 a^{4} + a^{3} + 7 a + 12\right)\cdot 19^{2} + \left(6 a^{5} + 14 a^{4} + 13 a^{3} + 8 a^{2} + 7 a + 10\right)\cdot 19^{3} + \left(15 a^{5} + 5 a^{3} + 10 a^{2} + 3 a + 1\right)\cdot 19^{4} + \left(2 a^{5} + 14 a^{4} + 8 a^{3} + 5 a^{2} + 15 a + 8\right)\cdot 19^{5} + \left(4 a^{4} + 7 a^{3} + 5 a^{2} + 4 a + 14\right)\cdot 19^{6} + \left(13 a^{5} + 4 a^{4} + 13 a^{3} + 6 a^{2} + 7 a + 10\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 2 }$ | $=$ | \( 18 a^{5} + 12 a^{4} + 17 a^{2} + 4 a + 9 + \left(12 a^{5} + 13 a^{4} + 9 a^{3} + 17 a^{2} + 13 a + 11\right)\cdot 19 + \left(a^{5} + 8 a^{4} + 14 a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 19^{2} + \left(3 a^{4} + 9 a^{3} + 16 a^{2} + 16 a + 10\right)\cdot 19^{3} + \left(6 a^{5} + 15 a^{4} + 16 a^{3} + 10 a^{2} + 17 a + 12\right)\cdot 19^{4} + \left(13 a^{5} + 13 a^{4} + 2 a^{3} + a^{2} + 12 a + 1\right)\cdot 19^{5} + \left(7 a^{5} + a^{4} + 4 a^{3} + 11 a^{2} + 14 a + 16\right)\cdot 19^{6} + \left(a^{5} + 4 a^{4} + 6 a^{3} + 14 a^{2} + 12 a + 5\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 3 }$ | $=$ | \( a^{5} + 2 a^{4} + 8 a^{3} + a^{2} + 12 a + 15 + \left(18 a^{5} + 2 a^{4} + 17 a^{3} + 17 a + 1\right)\cdot 19 + \left(15 a^{5} + 11 a^{4} + 16 a^{2} + 9 a + 1\right)\cdot 19^{2} + \left(12 a^{5} + 9 a^{4} + 16 a^{3} + 3 a^{2} + 4 a + 13\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 14 a^{3} + 8 a^{2} + 2 a + 6\right)\cdot 19^{4} + \left(7 a^{5} + 18 a^{4} + 13 a^{3} + 3 a^{2} + 8 a + 11\right)\cdot 19^{5} + \left(18 a^{5} + 6 a^{4} + 13 a^{3} + 5 a^{2} + 10 a + 13\right)\cdot 19^{6} + \left(7 a^{5} + 12 a^{4} + 5 a^{3} + 12 a^{2} + 14 a + 18\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 4 }$ | $=$ | \( 18 a^{4} + 11 a^{3} + 14 a^{2} + 3 a + 11 + \left(7 a^{5} + 6 a^{4} + 5 a^{3} + 11 a^{2} + 16 a + 11\right)\cdot 19 + \left(17 a^{5} + a^{4} + 11 a^{3} + 12 a^{2} + 5 a + 17\right)\cdot 19^{2} + \left(12 a^{5} + 14 a^{4} + 5 a^{3} + 5 a^{2} + a + 3\right)\cdot 19^{3} + \left(15 a^{5} + 18 a^{4} + 15 a^{3} + 3 a^{2} + 5 a\right)\cdot 19^{4} + \left(15 a^{5} + 16 a^{4} + 18 a^{3} + 10 a^{2} + 16 a + 9\right)\cdot 19^{5} + \left(18 a^{5} + 12 a^{3} + 10 a^{2} + 9 a + 16\right)\cdot 19^{6} + \left(3 a^{5} + 7 a^{4} + 13 a^{3} + 16 a^{2} + 14 a + 18\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 5 }$ | $=$ | \( 17 a^{5} + 14 a^{4} + 17 a^{3} + 18 a^{2} + 4 a + 16 + \left(12 a^{5} + 16 a^{4} + a^{3} + 10 a^{2} + 2 a + 1\right)\cdot 19 + \left(6 a^{5} + 4 a^{4} + 2 a^{3} + 6 a^{2} + 4 a + 18\right)\cdot 19^{2} + \left(9 a^{5} + 13 a^{4} + 9 a^{3} + 9 a^{2} + 14 a + 6\right)\cdot 19^{3} + \left(16 a^{5} + a^{3} + 2 a^{2} + 11 a + 7\right)\cdot 19^{4} + \left(13 a^{5} + 4 a^{4} + 11 a^{3} + 10 a^{2} + 7 a + 17\right)\cdot 19^{5} + \left(14 a^{5} + 10 a^{4} + 4 a^{3} + 10 a^{2} + 17 a + 14\right)\cdot 19^{6} + \left(18 a^{5} + 15 a^{4} + 13 a^{3} + 15 a^{2} + 4 a + 9\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 6 }$ | $=$ | \( 13 a^{5} + 6 a^{4} + 4 a^{3} + 16 a + 7 + \left(4 a^{5} + 17 a^{4} + 8 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 19 + \left(17 a^{5} + 17 a^{4} + 11 a^{3} + 7 a^{2} + 3\right)\cdot 19^{2} + \left(8 a^{5} + 13 a^{4} + 16 a^{3} + 4 a^{2} + 11 a\right)\cdot 19^{3} + \left(2 a^{5} + 12 a^{4} + 13 a^{3} + 10 a^{2} + 11 a + 8\right)\cdot 19^{4} + \left(7 a^{5} + 6 a^{4} + 6 a^{3} + 6 a^{2} + 3 a + 11\right)\cdot 19^{5} + \left(14 a^{5} + 8 a^{4} + 8 a^{3} + 11 a^{2} + 14 a + 14\right)\cdot 19^{6} + \left(18 a^{5} + 12 a^{4} + 18 a^{3} + 9 a^{2} + 16 a + 18\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 7 }$ | $=$ | \( 17 a^{5} + 8 a^{4} + 5 a^{3} + 6 a^{2} + 16 a + 14 + \left(3 a^{5} + 11 a^{4} + 11 a^{2} + 17 a + 4\right)\cdot 19 + \left(5 a^{5} + 18 a^{3} + 15 a^{2} + 2 a + 5\right)\cdot 19^{2} + \left(14 a^{5} + 10 a^{4} + 11 a^{3} + 18 a^{2} + 12 a + 6\right)\cdot 19^{3} + \left(6 a^{5} + 6 a^{4} + 10 a^{3} + 12 a^{2} + 2 a + 2\right)\cdot 19^{4} + \left(13 a^{5} + 4 a^{4} + 13 a^{3} + 5 a^{2} + 15 a + 5\right)\cdot 19^{5} + \left(6 a^{5} + 12 a^{4} + 17 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 19^{6} + \left(9 a^{5} + 14 a^{4} + 5 a^{3} + 17 a^{2} + 7 a + 18\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{5} + 2 a^{4} + 13 a^{3} + 7 a^{2} + 7 a + 11 + \left(12 a^{5} + 9 a^{4} + 5 a^{3} + 5 a^{2} + 7 a + 9\right)\cdot 19 + \left(12 a^{5} + 16 a^{4} + 14 a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 19^{2} + \left(15 a^{5} + 6 a^{3} + 12 a^{2} + 11 a + 14\right)\cdot 19^{3} + \left(8 a^{4} + 4 a^{2} + 10 a + 2\right)\cdot 19^{4} + \left(11 a^{3} + 16 a^{2} + 15 a + 8\right)\cdot 19^{5} + \left(12 a^{5} + 4 a^{4} + a^{3} + 17 a^{2} + 12 a + 16\right)\cdot 19^{6} + \left(16 a^{5} + 15 a^{4} + 14 a^{3} + 8 a^{2} + 13 a + 8\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 9 }$ | $=$ | \( a^{5} + 14 a^{4} + 6 a^{3} + 15 a^{2} + 15 a + 1 + \left(15 a^{5} + 3 a^{4} + 16 a^{3} + 17 a^{2} + 9 a + 18\right)\cdot 19 + \left(4 a^{5} + 15 a^{4} + 3 a^{3} + a^{2} + a + 3\right)\cdot 19^{2} + \left(17 a^{5} + 13 a^{4} + a^{3} + 5 a^{2} + 15 a + 4\right)\cdot 19^{3} + \left(a^{5} + 9 a^{4} + 7 a^{3} + 11 a^{2} + 6 a + 6\right)\cdot 19^{4} + \left(17 a^{5} + 4 a^{4} + 13 a^{3} + 10 a^{2} + 10 a + 8\right)\cdot 19^{5} + \left(2 a^{5} + 17 a^{4} + 16 a^{2} + 10 a + 14\right)\cdot 19^{6} + \left(18 a^{5} + 4 a^{4} + 5 a^{3} + 18 a^{2} + 7\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 10 }$ | $=$ | \( 13 a^{5} + a^{3} + 11 a^{2} + 5 a + 15 + \left(14 a^{5} + 18 a^{4} + 7 a^{3} + 18 a^{2} + 15 a + 16\right)\cdot 19 + \left(17 a^{5} + 12 a^{4} + 2 a^{3} + 9 a^{2} + 9 a + 14\right)\cdot 19^{2} + \left(11 a^{5} + 13 a^{4} + 9 a^{3} + 15 a^{2} + 16 a + 8\right)\cdot 19^{3} + \left(17 a^{5} + 10 a^{4} + 3 a^{3} + 10 a^{2} + 17 a + 5\right)\cdot 19^{4} + \left(18 a^{5} + 17 a^{4} + 8 a^{3} + 2 a^{2} + 10 a + 7\right)\cdot 19^{5} + \left(9 a^{5} + 14 a^{4} + 4 a^{3} + 15 a^{2} + 8 a + 17\right)\cdot 19^{6} + \left(4 a^{5} + 13 a^{4} + 13 a^{3} + 4 a^{2} + 12 a + 14\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 11 }$ | $=$ | \( 6 a^{5} + 10 a^{4} + a^{3} + 4 a^{2} + 11 a + 14 + \left(11 a^{5} + 8 a^{4} + 6 a^{3} + 15 a^{2} + 10 a + 2\right)\cdot 19 + \left(5 a^{4} + 4 a^{3} + 6 a^{2} + 13 a + 16\right)\cdot 19^{2} + \left(12 a^{5} + 16 a^{4} + 7 a^{3} + 7 a^{2} + 9 a + 3\right)\cdot 19^{3} + \left(13 a^{5} + 7 a^{4} + 10 a^{3} + 3 a^{2} + 18 a + 4\right)\cdot 19^{4} + \left(16 a^{5} + 17 a^{3} + 11 a^{2} + 3 a + 1\right)\cdot 19^{5} + \left(2 a^{5} + 10 a^{4} + 12 a^{3} + 16 a^{2} + 4 a + 18\right)\cdot 19^{6} + \left(a^{5} + 17 a^{4} + 2 a^{2} + 7 a + 17\right)\cdot 19^{7} +O(19^{8})\) |
$r_{ 12 }$ | $=$ | \( 10 a^{5} + 13 a^{4} + 3 a^{3} + 11 a^{2} + 15 a + 9 + \left(2 a^{5} + 10 a^{4} + 7 a^{3} + 17 a^{2} + 14 a + 8\right)\cdot 19 + \left(18 a^{5} + 3 a^{4} + 10 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 19^{2} + \left(10 a^{5} + 9 a^{4} + 7 a^{3} + 6 a^{2} + 13 a + 12\right)\cdot 19^{3} + \left(17 a^{5} + 12 a^{4} + 14 a^{3} + 6 a^{2} + 5 a + 18\right)\cdot 19^{4} + \left(5 a^{5} + 12 a^{4} + 7 a^{3} + 11 a^{2} + 13 a + 5\right)\cdot 19^{5} + \left(5 a^{5} + 3 a^{4} + 6 a^{3} + 8 a^{2} + 12 a + 7\right)\cdot 19^{6} + \left(11 a^{4} + 4 a^{3} + 5 a^{2} + a + 1\right)\cdot 19^{7} +O(19^{8})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,11)(2,9)(3,8)(4,12)(5,7)(6,10)$ | $-2$ |
$3$ | $2$ | $(1,5)(2,3)(4,6)(7,11)(8,9)(10,12)$ | $0$ |
$3$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $0$ |
$1$ | $3$ | $(1,10,9)(2,11,6)(3,7,4)(5,12,8)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,9,10)(2,6,11)(3,4,7)(5,8,12)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(3,4,7)(5,8,12)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(3,7,4)(5,12,8)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,9,10)(2,6,11)(3,7,4)(5,12,8)$ | $-1$ |
$1$ | $6$ | $(1,6,9,11,10,2)(3,5,4,8,7,12)$ | $-2 \zeta_{3}$ |
$1$ | $6$ | $(1,2,10,11,9,6)(3,12,7,8,4,5)$ | $2 \zeta_{3} + 2$ |
$2$ | $6$ | $(1,11)(2,9)(3,12,7,8,4,5)(6,10)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,11)(2,9)(3,5,4,8,7,12)(6,10)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,2,10,11,9,6)(3,5,4,8,7,12)$ | $1$ |
$3$ | $6$ | $(1,12,9,5,10,8)(2,7,6,3,11,4)$ | $0$ |
$3$ | $6$ | $(1,8,10,5,9,12)(2,4,11,3,6,7)$ | $0$ |
$3$ | $6$ | $(1,4,9,7,10,3)(2,5,6,8,11,12)$ | $0$ |
$3$ | $6$ | $(1,3,10,7,9,4)(2,12,11,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.