Basic invariants
Dimension: | $2$ |
Group: | $S_3\times C_3$ |
Conductor: | \(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \) |
Artin number field: | Galois closure of 6.0.33614000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3\times C_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.140.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 14 a + 28 + \left(5 a + 20\right)\cdot 41 + \left(22 a + 37\right)\cdot 41^{2} + \left(36 a + 27\right)\cdot 41^{3} + \left(2 a + 13\right)\cdot 41^{4} + \left(6 a + 31\right)\cdot 41^{5} +O(41^{6})\)
$r_{ 2 }$ |
$=$ |
\( 13 a + 15 + \left(6 a + 20\right)\cdot 41 + \left(33 a + 19\right)\cdot 41^{2} + \left(20 a + 16\right)\cdot 41^{3} + \left(34 a + 6\right)\cdot 41^{4} + \left(39 a + 23\right)\cdot 41^{5} +O(41^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 28 a + 13 + \left(34 a + 26\right)\cdot 41 + \left(7 a + 30\right)\cdot 41^{2} + \left(20 a + 4\right)\cdot 41^{3} + \left(6 a + 7\right)\cdot 41^{4} + \left(a + 26\right)\cdot 41^{5} +O(41^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 27 a + 29 + \left(35 a + 22\right)\cdot 41 + \left(18 a + 16\right)\cdot 41^{2} + \left(4 a + 33\right)\cdot 41^{3} + \left(38 a + 26\right)\cdot 41^{4} + \left(34 a + 5\right)\cdot 41^{5} +O(41^{6})\)
| $r_{ 5 }$ |
$=$ |
\( a + 39 + \left(40 a + 38\right)\cdot 41 + \left(29 a + 4\right)\cdot 41^{2} + \left(15 a + 32\right)\cdot 41^{3} + \left(9 a + 7\right)\cdot 41^{4} + \left(7 a + 12\right)\cdot 41^{5} +O(41^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 40 a + 1 + 35\cdot 41 + \left(11 a + 13\right)\cdot 41^{2} + \left(25 a + 8\right)\cdot 41^{3} + \left(31 a + 20\right)\cdot 41^{4} + \left(33 a + 24\right)\cdot 41^{5} +O(41^{6})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ | $0$ |
$1$ | $3$ | $(1,3,6)(2,5,4)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
$1$ | $3$ | $(1,6,3)(2,4,5)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
$2$ | $3$ | $(1,3,6)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,6,3)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,6,3)(2,5,4)$ | $-1$ | $-1$ |
$3$ | $6$ | $(1,5,3,4,6,2)$ | $0$ | $0$ |
$3$ | $6$ | $(1,2,6,4,3,5)$ | $0$ | $0$ |