Basic invariants
Dimension: | $2$ |
Group: | $C_6\times S_3$ |
Conductor: | \(2888\)\(\medspace = 2^{3} \cdot 19^{2} \) |
Artin stem field: | Galois closure of 12.0.4452139149819904.6 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6\times S_3$ |
Parity: | odd |
Determinant: | 1.152.6t1.c.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.2888.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 4 x^{11} + 10 x^{10} - 8 x^{9} + 16 x^{8} - 26 x^{7} + 74 x^{6} - 12 x^{5} + 72 x^{4} - 4 x^{3} + 207 x^{2} + 154 x + 139 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 3 a^{5} + 3 a^{4} + 10 a^{3} + 5 a^{2} + 11 a + 1 + \left(8 a^{5} + a^{4} + 2 a^{3} + 7 a^{2} + 7 a + 8\right)\cdot 13 + \left(7 a^{5} + 12 a^{4} + 5 a^{3} + 2 a^{2} + 12 a + 9\right)\cdot 13^{2} + \left(3 a^{5} + 8 a^{4} + 11 a^{3} + a^{2} + 12\right)\cdot 13^{3} + \left(4 a^{5} + 7 a^{4} + 3 a^{3} + 12 a^{2} + 6 a + 12\right)\cdot 13^{4} + \left(7 a^{5} + 7 a^{3} + 8 a^{2} + 9 a + 11\right)\cdot 13^{5} + \left(7 a^{5} + 12 a^{4} + 2 a^{3}\right)\cdot 13^{6} + \left(11 a^{5} + 12 a^{4} + 12 a^{3} + 5 a^{2} + 6 a + 2\right)\cdot 13^{7} + \left(3 a^{5} + 10 a^{4} + 4 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 13^{8} +O(13^{9})\)
$r_{ 2 }$ |
$=$ |
\( 9 a^{5} + 4 a^{2} + 6 a + 10 + \left(4 a^{5} + 11 a^{2} + 11 a + 4\right)\cdot 13 + \left(7 a^{5} + 11 a^{4} + 8 a^{3} + 9 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(4 a^{5} + 12 a^{4} + a^{3} + 2 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(4 a^{5} + 8 a^{4} + 5 a^{3} + 4 a^{2} + 10 a + 9\right)\cdot 13^{4} + \left(11 a^{5} + 5 a^{4} + a^{3} + 2 a^{2} + 3 a + 6\right)\cdot 13^{5} + \left(12 a^{5} + 11 a^{4} + 12 a^{3} + 12 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(2 a^{5} + 7 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 11\right)\cdot 13^{7} + \left(a^{5} + 8 a^{4} + 6 a^{3} + 2 a^{2} + 11 a + 7\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 5 a^{4} + 4 a^{3} + 4 a^{2} + 8 a + 2 + \left(8 a^{5} + 3 a^{4} + 11 a^{3} + 3 a^{2} + 10 a\right)\cdot 13 + \left(a^{5} + 12 a^{4} + a^{3} + a^{2} + 4 a + 1\right)\cdot 13^{2} + \left(5 a^{5} + 5 a^{4} + a^{3} + 10 a^{2} + 7\right)\cdot 13^{3} + \left(12 a^{5} + 2 a^{4} + 4 a^{3} + 2 a^{2} + 12 a\right)\cdot 13^{4} + \left(10 a^{5} + 7 a^{4} + 2 a^{3} + 11 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(8 a^{5} + a^{4} + 11 a^{3} + 10 a + 12\right)\cdot 13^{6} + \left(12 a^{5} + a^{4} + 2 a^{3} + 12 a^{2} + 2 a + 12\right)\cdot 13^{7} + \left(12 a^{5} + 11 a^{4} + 12 a^{3} + 6 a^{2} + 10 a + 3\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{5} + 9 a^{4} + 7 a^{3} + 10 a^{2} + 11 a + \left(2 a^{5} + 3 a^{4} + 8 a^{3} + 2 a^{2} + 5 a + 10\right)\cdot 13 + \left(7 a^{5} + 11 a^{4} + 4 a^{3} + 2 a^{2} + 3 a + 4\right)\cdot 13^{2} + \left(a^{5} + 6 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(12 a^{5} + 8 a^{4} + 6 a^{3} + a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(5 a^{5} + 4 a^{4} + 11 a^{3} + 9 a^{2} + 10 a + 12\right)\cdot 13^{5} + \left(9 a^{5} + 10 a^{4} + a^{3} + 12 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(2 a^{4} + 2 a^{3} + 6 a^{2} + 9 a + 9\right)\cdot 13^{7} + \left(12 a^{5} + 10 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a^{5} + 4 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 1 + \left(5 a^{5} + 2 a^{4} + 4 a^{3} + 8 a^{2} + 3\right)\cdot 13 + \left(a^{5} + 5 a^{4} + 5 a^{3} + 10 a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(7 a^{5} + 9 a^{4} + 7 a^{3} + 11 a^{2} + 1\right)\cdot 13^{3} + \left(11 a^{5} + 2 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{4} + \left(7 a^{5} + 8 a^{4} + 10 a^{3} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{5} + 7 a^{4} + 8 a^{3} + 5 a^{2} + 9 a + 11\right)\cdot 13^{6} + \left(4 a^{5} + 9 a^{4} + 7 a^{3} + 9 a + 1\right)\cdot 13^{7} + \left(2 a^{5} + 7 a^{4} + 2 a^{3} + 6 a^{2} + 4 a + 7\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 11 a^{5} + 6 a^{4} + 3 a^{3} + 8 a^{2} + 4 a + 5 + \left(12 a^{5} + 2 a^{4} + 6 a^{3} + 12 a^{2} + 11 a + 7\right)\cdot 13 + \left(a^{5} + 10 a^{3} + 6 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(10 a^{5} + 10 a^{4} + 2 a^{3} + 12 a^{2} + 9\right)\cdot 13^{3} + \left(5 a^{4} + 8 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(12 a^{4} + a^{3} + 2 a + 8\right)\cdot 13^{5} + \left(3 a^{5} + 3 a^{4} + 2 a^{3} + 5 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(4 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + a + 12\right)\cdot 13^{7} + \left(6 a^{5} + 6 a^{4} + 4 a^{3} + 3 a^{2} + 9 a + 11\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 7 }$ |
$=$ |
\( 12 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + a + 5 + \left(8 a^{5} + 4 a^{4} + 9 a^{3} + 8 a^{2} + 6 a + 5\right)\cdot 13 + \left(2 a^{5} + 10 a^{4} + 5 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 13^{2} + \left(12 a^{5} + a^{3} + 2 a + 10\right)\cdot 13^{3} + \left(10 a^{5} + 12 a^{4} + 9 a^{3} + a^{2} + 2 a + 8\right)\cdot 13^{4} + \left(12 a^{5} + 5 a^{4} + 9 a^{3} + a^{2} + 6 a + 9\right)\cdot 13^{5} + \left(8 a^{5} + 9 a^{4} + 10 a^{3} + 8 a^{2} + 8 a + 7\right)\cdot 13^{6} + \left(6 a^{5} + 2 a^{4} + 3 a^{3} + 5 a^{2} + 4 a + 9\right)\cdot 13^{7} + \left(8 a^{4} + 2 a^{3} + 5 a^{2} + 11 a + 8\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 8 }$ |
$=$ |
\( 12 a^{5} + 2 a^{4} + 5 a^{3} + a^{2} + 3 a + 10 + \left(10 a^{5} + 2 a^{4} + 9 a^{2} + 3\right)\cdot 13 + \left(2 a^{5} + 7 a^{4} + 7 a^{3} + 5 a + 12\right)\cdot 13^{2} + \left(4 a^{5} + 6 a^{4} + 2 a^{3} + 3 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(9 a^{5} + 3 a^{4} + 6 a^{3} + 5 a^{2} + 5 a + 9\right)\cdot 13^{4} + \left(3 a^{5} + 6 a^{4} + 3 a^{3} + 9 a^{2} + 5\right)\cdot 13^{5} + \left(2 a^{5} + 7 a^{4} + 7 a^{3} + 6 a + 5\right)\cdot 13^{6} + \left(9 a^{5} + 8 a^{4} + 2 a^{3} + 10 a + 10\right)\cdot 13^{7} + \left(9 a^{5} + 4 a^{4} + 9 a^{3} + 5 a^{2} + 1\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 9 }$ |
$=$ |
\( 6 a^{5} + 3 a^{4} + 3 a^{3} + 3 a^{2} + 2 a + 7 + \left(4 a^{5} + 4 a^{4} + 4 a^{3} + 3 a^{2} + 9 a + 3\right)\cdot 13 + \left(9 a^{5} + 11 a^{4} + 4 a^{3} + 9 a^{2} + 10 a + 12\right)\cdot 13^{2} + \left(11 a^{5} + 9 a^{4} + a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13^{3} + \left(12 a^{5} + 11 a^{3} + 7 a + 6\right)\cdot 13^{4} + \left(9 a^{5} + 7 a^{4} + 2 a^{3} + 11 a^{2} + 10 a + 4\right)\cdot 13^{5} + \left(8 a^{5} + 9 a^{4} + 2 a^{3} + 12 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(a^{5} + 10 a^{4} + 12 a^{3} + 6 a^{2} + 11 a + 11\right)\cdot 13^{7} + \left(11 a^{5} + 3 a^{4} + 7 a^{3} + 2 a^{2} + 3 a + 9\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 10 }$ |
$=$ |
\( a^{5} + 12 a^{4} + 12 a^{3} + a^{2} + 11 a + 5 + \left(12 a^{5} + 12 a^{4} + 6 a^{3} + 8 a^{2} + 11 a + 5\right)\cdot 13 + \left(10 a^{5} + 12 a^{4} + 7 a^{3} + 4 a^{2} + 11 a + 7\right)\cdot 13^{2} + \left(12 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(11 a^{5} + 6 a^{4} + 4 a^{3} + 8 a^{2} + 3\right)\cdot 13^{4} + \left(8 a^{5} + 12 a^{4} + 10 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(12 a^{5} + 4 a^{4} + 7 a^{3} + 7 a^{2} + 12 a + 8\right)\cdot 13^{6} + \left(7 a^{5} + 9 a^{4} + 2 a^{2} + 10 a + 3\right)\cdot 13^{7} + \left(2 a^{5} + 9 a^{4} + 11 a^{3} + a^{2} + 4 a + 12\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 11 }$ |
$=$ |
\( 6 a^{5} + 6 a^{4} + 4 a^{3} + 11 a + 8 + \left(6 a^{5} + 12 a^{4} + 3 a^{3} + 12\right)\cdot 13 + \left(8 a^{5} + 11 a^{4} + 2 a^{3} + 11 a^{2} + 3 a\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + 6 a^{2} + 6 a\right)\cdot 13^{3} + \left(4 a^{5} + 4 a^{4} + 5 a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(8 a^{5} + 12 a^{4} + 11 a^{3} + 9 a^{2} + 4 a + 12\right)\cdot 13^{5} + \left(6 a^{5} + 4 a^{4} + 7 a^{3} + 12 a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(11 a^{5} + 5 a^{4} + 9 a^{2} + 7 a + 11\right)\cdot 13^{7} + \left(10 a^{5} + 6 a^{4} + 8 a^{3} + 3 a^{2} + 12 a + 9\right)\cdot 13^{8} +O(13^{9})\)
| $r_{ 12 }$ |
$=$ |
\( 5 a^{5} + 3 a^{4} + 10 a^{3} + 7 a^{2} + a + 2 + \left(6 a^{5} + 2 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 1\right)\cdot 13 + \left(3 a^{5} + 11 a^{4} + 2 a^{3} + 4 a + 12\right)\cdot 13^{2} + \left(4 a^{5} + 12 a^{4} + 6 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 13^{3} + \left(9 a^{5} + 6 a^{3} + 10 a^{2} + 10 a + 5\right)\cdot 13^{4} + \left(3 a^{5} + 8 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 5\right)\cdot 13^{5} + \left(4 a^{5} + 7 a^{4} + 4 a^{3} + 12 a^{2} + 12 a + 9\right)\cdot 13^{6} + \left(4 a^{5} + 11 a^{4} + 9 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 13^{7} + \left(4 a^{5} + 12 a^{4} + 11 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 13^{8} +O(13^{9})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,11)(4,6)(5,8)(7,9)(10,12)$ | $-2$ |
$3$ | $2$ | $(1,4)(2,10)(3,6)(5,7)(8,9)(11,12)$ | $0$ |
$3$ | $2$ | $(1,6)(2,12)(3,4)(5,9)(7,8)(10,11)$ | $0$ |
$1$ | $3$ | $(1,8,12)(2,6,7)(3,5,10)(4,9,11)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,12,8)(2,7,6)(3,10,5)(4,11,9)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(2,6,7)(4,9,11)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(2,7,6)(4,11,9)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,12,8)(2,6,7)(3,10,5)(4,9,11)$ | $-1$ |
$1$ | $6$ | $(1,10,8,3,12,5)(2,9,6,11,7,4)$ | $2 \zeta_{3} + 2$ |
$1$ | $6$ | $(1,5,12,3,8,10)(2,4,7,11,6,9)$ | $-2 \zeta_{3}$ |
$2$ | $6$ | $(1,3)(2,4,7,11,6,9)(5,8)(10,12)$ | $-\zeta_{3} - 1$ |
$2$ | $6$ | $(1,3)(2,9,6,11,7,4)(5,8)(10,12)$ | $\zeta_{3}$ |
$2$ | $6$ | $(1,5,12,3,8,10)(2,9,6,11,7,4)$ | $1$ |
$3$ | $6$ | $(1,2,8,6,12,7)(3,11,5,4,10,9)$ | $0$ |
$3$ | $6$ | $(1,7,12,6,8,2)(3,9,10,4,5,11)$ | $0$ |
$3$ | $6$ | $(1,9,8,11,12,4)(2,10,6,3,7,5)$ | $0$ |
$3$ | $6$ | $(1,4,12,11,8,9)(2,5,7,3,6,10)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.