Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.21296.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 6.0.21296.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + 1 + \left(4 a + 6\right)\cdot 7 + 5 a\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} +O(7^{5})\)
$r_{ 2 }$ |
$=$ |
\( 2 a + 4 + \left(4 a + 3\right)\cdot 7 + \left(2 a + 2\right)\cdot 7^{2} + \left(5 a + 5\right)\cdot 7^{3} + \left(a + 4\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 3 a + 3 + 6\cdot 7 + 3\cdot 7^{2} + 3 a\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a + 2 + \left(2 a + 2\right)\cdot 7 + \left(a + 2\right)\cdot 7^{2} + 4 a\cdot 7^{3} + \left(3 a + 5\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a + 6 + \left(2 a + 5\right)\cdot 7 + 4 a\cdot 7^{2} + \left(a + 1\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 4 a + 6 + \left(6 a + 3\right)\cdot 7 + \left(6 a + 3\right)\cdot 7^{2} + \left(3 a + 3\right)\cdot 7^{3} + 4\cdot 7^{4} +O(7^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
$2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ |