Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(1560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
Artin number field: | Galois closure of 12.0.85282689024000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.20280.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{6} + 17x^{3} + 17x^{2} + 6x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 12 a^{5} + 6 a^{4} + 8 a^{3} + 3 a^{2} + 13 a + 9 + \left(5 a^{5} + 15 a^{4} + 3 a^{3} + 6 a^{2} + 6 a\right)\cdot 19 + \left(3 a^{5} + 12 a^{4} + 15 a^{3} + 10 a^{2} + 5 a + 13\right)\cdot 19^{2} + \left(10 a^{5} + 17 a^{4} + 17 a^{3} + 13 a^{2} + 14 a + 18\right)\cdot 19^{3} + \left(13 a^{5} + 12 a^{4} + 15 a^{3} + a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(15 a^{5} + 5 a^{4} + a^{3} + 17 a^{2} + 9 a + 7\right)\cdot 19^{5} + \left(7 a^{5} + 4 a^{4} + 12 a^{3} + 3 a^{2} + 6 a + 18\right)\cdot 19^{6} + \left(15 a^{5} + 5 a^{4} + 7 a^{3} + 18 a^{2} + 15 a + 9\right)\cdot 19^{7} + \left(8 a^{5} + 14 a^{4} + 18 a^{2} + 15 a\right)\cdot 19^{8} + \left(13 a^{5} + 15 a^{3} + 5 a^{2} + 8 a + 13\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 8 a^{5} + 7 a^{4} + 8 a^{3} + 6 a^{2} + 12 a + 13 + \left(2 a^{5} + 4 a^{4} + 7 a^{3} + 4 a^{2} + 16 a + 14\right)\cdot 19 + \left(9 a^{5} + 2 a^{4} + 9 a^{3} + 12 a^{2} + 15 a + 18\right)\cdot 19^{2} + \left(16 a^{5} + 11 a^{4} + 8 a^{3} + 13 a^{2} + 12 a + 13\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 10 a^{3} + 15 a^{2} + 8 a + 11\right)\cdot 19^{4} + \left(a^{5} + 2 a^{4} + 17 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 19^{5} + \left(16 a^{5} + 13 a^{4} + 10 a^{3} + 12 a^{2} + a + 4\right)\cdot 19^{6} + \left(a^{5} + 6 a^{4} + 4 a^{3} + 18 a^{2} + 16 a + 18\right)\cdot 19^{7} + \left(3 a^{5} + 10 a^{4} + 8 a^{3} + 4 a^{2} + 10 a + 9\right)\cdot 19^{8} + \left(18 a^{5} + 8 a^{4} + 13 a^{3} + 2 a^{2} + 4 a\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 18 a^{5} + 10 a^{4} + 5 a^{3} + 16 a^{2} + 4 a + 5 + \left(5 a^{5} + 14 a^{4} + 15 a^{3} + 15 a^{2} + 9 a + 8\right)\cdot 19 + \left(10 a^{5} + 3 a^{4} + 14 a^{3} + 11 a^{2} + 14 a + 18\right)\cdot 19^{2} + \left(12 a^{5} + 17 a^{4} + 5 a^{3} + 4 a^{2} + a + 17\right)\cdot 19^{3} + \left(15 a^{5} + a^{4} + 16 a^{3} + 11 a^{2} + 3 a\right)\cdot 19^{4} + \left(16 a^{5} + 13 a^{4} + 17 a^{2} + a + 13\right)\cdot 19^{5} + \left(12 a^{5} + 3 a^{4} + 11 a^{3} + 10 a^{2} + 6 a + 5\right)\cdot 19^{6} + \left(a^{5} + 16 a^{4} + 7 a^{2} + 16 a + 18\right)\cdot 19^{7} + \left(7 a^{5} + 10 a^{4} + 6 a^{3} + 8 a^{2} + 4 a\right)\cdot 19^{8} + \left(7 a^{4} + 11 a^{3} + 17 a^{2} + 16 a + 6\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 18 a^{5} + a^{4} + 16 a^{3} + 12 a^{2} + 9 a + 13 + \left(3 a^{5} + 4 a^{4} + 9 a^{3} + 18 a^{2} + 14 a + 18\right)\cdot 19 + \left(4 a^{5} + 6 a^{3} + 16 a^{2} + 18 a + 4\right)\cdot 19^{2} + \left(4 a^{5} + 2 a^{4} + 10 a^{3} + 8 a^{2} + 10 a + 15\right)\cdot 19^{3} + \left(2 a^{5} + 17 a^{4} + 3 a^{3} + 17 a^{2} + a\right)\cdot 19^{4} + \left(10 a^{5} + 12 a^{4} + 14 a^{3} + 16 a^{2} + 5 a + 9\right)\cdot 19^{5} + \left(9 a^{5} + 9 a^{4} + 17 a^{3} + 9 a^{2} + 2 a + 5\right)\cdot 19^{6} + \left(2 a^{5} + 6 a^{4} + 13 a^{3} + 7 a^{2} + 11 a + 3\right)\cdot 19^{7} + \left(8 a^{5} + 10 a^{4} + 3 a^{3} + 7 a^{2} + 18 a + 15\right)\cdot 19^{8} + \left(11 a^{5} + 8 a^{4} + 7 a^{3} + 7 a^{2} + 15 a + 9\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( 5 a^{4} + 7 a^{3} + 3 a^{2} + 14 a + 9 + \left(13 a^{5} + 4 a^{4} + 3 a^{3} + a + 14\right)\cdot 19 + \left(15 a^{5} + 2 a^{4} + 5 a^{3} + 17 a^{2} + 15 a + 16\right)\cdot 19^{2} + \left(a^{5} + 11 a^{4} + 10 a^{2} + 13 a + 1\right)\cdot 19^{3} + \left(16 a^{5} + 5 a^{4} + 8 a^{3} + a + 6\right)\cdot 19^{4} + \left(5 a^{4} + 18 a^{3} + 5 a^{2} + 6 a + 4\right)\cdot 19^{5} + \left(8 a^{5} + 10 a^{4} + 15 a^{3} + 9 a^{2} + 9 a + 18\right)\cdot 19^{6} + \left(11 a^{5} + 4 a^{4} + 2 a^{3} + 8 a^{2} + 18 a\right)\cdot 19^{7} + \left(8 a^{5} + 4 a^{4} + a^{3} + 3 a^{2} + 17 a + 15\right)\cdot 19^{8} + \left(a^{5} + 16 a^{4} + 14 a^{2} + 9 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( a^{5} + 7 a^{4} + 12 a^{3} + 14 a^{2} + \left(2 a^{5} + 13 a^{4} + 16 a^{3} + a + 11\right)\cdot 19 + \left(9 a^{5} + 17 a^{4} + 9 a^{3} + 8 a^{2} + 10 a + 15\right)\cdot 19^{2} + \left(12 a^{5} + 5 a^{4} + 5 a^{3} + 16 a^{2} + a + 18\right)\cdot 19^{3} + \left(5 a^{5} + 6 a^{4} + 13 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 19^{4} + \left(6 a^{5} + 6 a^{4} + 3 a^{3} + 9 a^{2} + 17 a + 16\right)\cdot 19^{5} + \left(a^{5} + 13 a^{4} + 14 a^{3} + 18 a^{2} + 14 a + 16\right)\cdot 19^{6} + \left(15 a^{5} + 12 a^{4} + a^{3} + 6 a^{2} + 11 a + 16\right)\cdot 19^{7} + \left(4 a^{5} + 14 a^{4} + 5 a^{3} + 11 a^{2} + 11 a + 17\right)\cdot 19^{8} + \left(2 a^{5} + 11 a^{4} + 3 a^{3} + 15 a^{2} + 3 a + 1\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 7 }$ | $=$ | \( 12 a^{5} + 2 a^{4} + 6 a^{3} + 16 a^{2} + 3 a + 10 + \left(10 a^{5} + 15 a^{3} + 17 a^{2} + 12 a + 17\right)\cdot 19 + \left(18 a^{5} + 13 a^{4} + 13 a^{3} + 13 a^{2} + 7 a + 3\right)\cdot 19^{2} + \left(8 a^{5} + 9 a^{4} + 4 a^{3} + 4 a + 4\right)\cdot 19^{3} + \left(3 a^{5} + 6 a^{4} + 11 a^{3} + 11 a^{2} + 7 a + 6\right)\cdot 19^{4} + \left(3 a^{4} + 11 a^{2} + 12 a + 13\right)\cdot 19^{5} + \left(9 a^{5} + 2 a^{4} + 16 a^{3} + 14 a^{2} + 11 a + 14\right)\cdot 19^{6} + \left(15 a^{5} + 15 a^{4} + 13 a^{3} + 11 a^{2} + 5 a + 16\right)\cdot 19^{7} + \left(8 a^{5} + 16 a^{4} + 4 a^{3} + 5 a^{2} + 3 a + 8\right)\cdot 19^{8} + \left(2 a^{4} + 13 a^{3} + 18 a^{2} + 17 a + 1\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 8 }$ | $=$ | \( 12 a^{5} + 7 a^{4} + 16 a^{3} + 7 a + 6 + \left(18 a^{5} + 2 a^{3} + a + 8\right)\cdot 19 + \left(15 a^{5} + 12 a^{4} + 8 a^{3} + 12 a^{2} + 1\right)\cdot 19^{2} + \left(18 a^{5} + 12 a^{4} + 6 a^{3} + 13 a^{2} + 4 a\right)\cdot 19^{3} + \left(9 a^{5} + 14 a^{4} + 18 a^{3} + 14 a^{2} + 18 a + 11\right)\cdot 19^{4} + \left(3 a^{5} + 17 a^{4} + 18 a^{3} + 17 a^{2} + 4\right)\cdot 19^{5} + \left(12 a^{5} + 14 a^{4} + 8 a^{3} + 18 a^{2} + 17 a + 17\right)\cdot 19^{6} + \left(7 a^{5} + 14 a^{4} + 13 a^{3} + 9 a^{2} + 2 a + 11\right)\cdot 19^{7} + \left(13 a^{5} + 5 a^{4} + 11 a^{3} + 18 a^{2} + 10 a + 10\right)\cdot 19^{8} + \left(11 a^{4} + 7 a^{3} + 15 a^{2} + 18\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 9 }$ | $=$ | \( 14 a^{5} + 6 a^{4} + 14 a^{3} + 16 a^{2} + 18 a + 13 + \left(13 a^{5} + 3 a^{4} + 12 a^{3} + 12 a^{2} + 10 a + 12\right)\cdot 19 + \left(18 a^{5} + 13 a^{4} + 14 a^{3} + 15 a^{2} + 13 a + 7\right)\cdot 19^{2} + \left(8 a^{5} + 7 a^{4} + 13 a^{3} + 10 a^{2} + 17\right)\cdot 19^{3} + \left(14 a^{5} + 10 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 12\right)\cdot 19^{4} + \left(18 a^{5} + 14 a^{4} + 17 a^{3} + 3 a^{2} + 8 a\right)\cdot 19^{5} + \left(17 a^{5} + 18 a^{4} + 16 a^{3} + 15 a^{2} + 14 a + 8\right)\cdot 19^{6} + \left(14 a^{5} + 17 a^{4} + 16 a^{3} + 9 a^{2} + 12\right)\cdot 19^{7} + \left(15 a^{5} + 17 a^{4} + 6 a^{3} + 12 a + 8\right)\cdot 19^{8} + \left(4 a^{5} + 6 a^{4} + 15 a^{3} + 16 a^{2} + 9 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 10 }$ | $=$ | \( 18 a^{5} + 7 a^{4} + 2 a^{2} + 5 a + 2 + \left(3 a^{5} + a^{4} + 18 a^{3} + 18 a^{2} + 16 a + 10\right)\cdot 19 + \left(13 a^{5} + 18 a^{4} + 3 a^{3} + 12 a^{2} + 12 a + 2\right)\cdot 19^{2} + \left(4 a^{5} + a^{4} + 13 a^{3} + 10 a^{2} + 3 a\right)\cdot 19^{3} + \left(16 a^{5} + 7 a^{4} + 16 a^{3} + 5 a^{2} + 8 a + 6\right)\cdot 19^{4} + \left(11 a^{5} + 7 a^{4} + 15 a^{3} + 4 a^{2} + 14 a + 4\right)\cdot 19^{5} + \left(9 a^{5} + 14 a^{4} + 7 a^{3} + 10 a^{2} + 13 a + 16\right)\cdot 19^{6} + \left(11 a^{5} + a^{4} + 14 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 19^{7} + \left(5 a^{5} + 12 a^{3} + 4 a^{2} + 8 a + 4\right)\cdot 19^{8} + \left(15 a^{5} + 10 a^{4} + 15 a^{3} + 8 a^{2} + 5 a + 13\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 11 }$ | $=$ | \( 13 a^{5} + 5 a^{4} + 14 a^{3} + 12 a^{2} + 12 a + 10 + \left(2 a^{5} + 8 a^{4} + 8 a^{3} + a^{2} + 11 a + 9\right)\cdot 19 + \left(17 a^{5} + 14 a^{4} + 16 a^{3} + 9 a + 14\right)\cdot 19^{2} + \left(10 a^{5} + 13 a^{4} + 16 a^{3} + 11 a + 18\right)\cdot 19^{3} + \left(4 a^{5} + 3 a^{4} + 16 a^{3} + 10 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(8 a^{5} + 10 a^{4} + 11 a^{3} + 2 a^{2} + a + 3\right)\cdot 19^{5} + \left(10 a^{5} + a^{3} + 3 a^{2} + 3 a + 13\right)\cdot 19^{6} + \left(2 a^{5} + 4 a^{4} + 14 a^{3} + 15 a^{2} + 2 a + 14\right)\cdot 19^{7} + \left(2 a^{5} + 12 a^{4} + 3 a^{3} + 14 a^{2} + 16 a + 12\right)\cdot 19^{8} + \left(a^{5} + 7 a^{4} + 8 a^{3} + 17 a^{2} + 16\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 12 }$ | $=$ | \( 7 a^{5} + 13 a^{4} + 8 a^{3} + 14 a^{2} + 17 a + 7 + \left(12 a^{5} + 6 a^{4} + 17 a^{2} + 11 a + 7\right)\cdot 19 + \left(16 a^{5} + 4 a^{4} + 15 a^{3} + a^{2} + 9 a + 15\right)\cdot 19^{2} + \left(3 a^{5} + 3 a^{4} + 10 a^{3} + 10 a^{2} + 15 a + 5\right)\cdot 19^{3} + \left(12 a^{5} + 17 a^{4} + 17 a^{3} + 10 a^{2} + 14 a + 12\right)\cdot 19^{4} + \left(14 a^{4} + 11 a^{3} + 18 a^{2} + 12 a + 12\right)\cdot 19^{5} + \left(18 a^{5} + 8 a^{4} + 18 a^{3} + 5 a^{2} + 13 a + 13\right)\cdot 19^{6} + \left(13 a^{5} + 8 a^{4} + 9 a^{3} + 15 a^{2} + 5 a + 4\right)\cdot 19^{7} + \left(8 a^{5} + 15 a^{4} + 11 a^{3} + 15 a^{2} + 3 a + 9\right)\cdot 19^{8} + \left(6 a^{5} + 2 a^{4} + 3 a^{3} + 12 a^{2} + 2 a + 3\right)\cdot 19^{9} +O(19^{10})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,11)(2,6)(3,10)(4,9)(5,7)(8,12)$ | $-2$ | $-2$ |
$3$ | $2$ | $(1,6)(2,11)(3,12)(4,7)(5,9)(8,10)$ | $0$ | $0$ |
$3$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $0$ | $0$ |
$1$ | $3$ | $(1,9,8)(2,7,3)(4,12,11)(5,10,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,8,9)(2,3,7)(4,11,12)(5,6,10)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(1,9,8)(4,12,11)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,8,9)(4,11,12)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,9,8)(2,3,7)(4,12,11)(5,6,10)$ | $-1$ | $-1$ |
$1$ | $6$ | $(1,4,8,11,9,12)(2,5,3,6,7,10)$ | $2 \zeta_{3} + 2$ | $-2 \zeta_{3}$ |
$1$ | $6$ | $(1,12,9,11,8,4)(2,10,7,6,3,5)$ | $-2 \zeta_{3}$ | $2 \zeta_{3} + 2$ |
$2$ | $6$ | $(1,12,9,11,8,4)(2,6)(3,10)(5,7)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,4,8,11,9,12)(2,6)(3,10)(5,7)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,12,9,11,8,4)(2,5,3,6,7,10)$ | $1$ | $1$ |
$3$ | $6$ | $(1,10,9,6,8,5)(2,12,7,11,3,4)$ | $0$ | $0$ |
$3$ | $6$ | $(1,5,8,6,9,10)(2,4,3,11,7,12)$ | $0$ | $0$ |
$3$ | $6$ | $(1,2,9,7,8,3)(4,5,12,10,11,6)$ | $0$ | $0$ |
$3$ | $6$ | $(1,3,8,7,9,2)(4,6,11,10,12,5)$ | $0$ | $0$ |