Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(351\)\(\medspace = 3^{3} \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 6.0.4804839.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Determinant: | 1.39.2t1.a.a |
Projective image: | $S_3$ |
Projective field: | Galois closure of 6.0.4804839.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 9x^{4} - 13x^{3} + 9x^{2} - 3x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( a + 4 + \left(4 a + 2\right)\cdot 7 + \left(5 a + 1\right)\cdot 7^{2} + \left(2 a + 2\right)\cdot 7^{3} + \left(2 a + 4\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 4 + \left(2 a + 4\right)\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(4 a + 4\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 5 + \left(2 a + 5\right)\cdot 7 + \left(a + 2\right)\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + \left(4 a + 3\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a + 2 + 3 a\cdot 7 + 3 a\cdot 7^{2} + \left(4 a + 3\right)\cdot 7^{3} + \left(5 a + 6\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 5 }$ | $=$ | \( a + 3 + \left(4 a + 1\right)\cdot 7 + \left(5 a + 4\right)\cdot 7^{2} + 2 a\cdot 7^{3} + \left(2 a + 3\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 6 }$ | $=$ | \( 3 a + 6 + \left(3 a + 6\right)\cdot 7 + \left(3 a + 6\right)\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + a\cdot 7^{4} +O(7^{5})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.