Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(6223\)\(\medspace = 7^{2} \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.304927.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.127.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.790321.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{3} - 7x^{2} + 6x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 12 a + 6 + 8 a\cdot 13 + \left(9 a + 10\right)\cdot 13^{2} + \left(a + 7\right)\cdot 13^{3} + \left(3 a + 6\right)\cdot 13^{4} + \left(7 a + 12\right)\cdot 13^{5} + \left(11 a + 2\right)\cdot 13^{6} + \left(11 a + 11\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 2 }$ | $=$ | \( 9 a + 6 + \left(9 a + 12\right)\cdot 13 + \left(7 a + 11\right)\cdot 13^{2} + \left(9 a + 5\right)\cdot 13^{3} + \left(10 a + 11\right)\cdot 13^{4} + \left(6 a + 8\right)\cdot 13^{5} + \left(11 a + 10\right)\cdot 13^{6} + \left(12 a + 2\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 3 }$ | $=$ | \( a + 5 + \left(4 a + 10\right)\cdot 13 + \left(3 a + 10\right)\cdot 13^{2} + \left(11 a + 12\right)\cdot 13^{3} + \left(9 a + 7\right)\cdot 13^{4} + \left(5 a + 3\right)\cdot 13^{5} + \left(a + 7\right)\cdot 13^{6} + \left(a + 11\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 4 }$ | $=$ | \( 12 + 3\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{4} + 3\cdot 13^{5} + 5\cdot 13^{6} + 6\cdot 13^{7} +O(13^{8})\) |
$r_{ 5 }$ | $=$ | \( 4 a + 2 + 3 a\cdot 13 + \left(5 a + 10\right)\cdot 13^{2} + \left(3 a + 7\right)\cdot 13^{3} + \left(2 a + 12\right)\cdot 13^{4} + \left(6 a + 4\right)\cdot 13^{5} + \left(a + 2\right)\cdot 13^{6} + 4\cdot 13^{7} +O(13^{8})\) |
$r_{ 6 }$ | $=$ | \( 10 + 11\cdot 13 + 11\cdot 13^{2} + 3\cdot 13^{3} + 10\cdot 13^{4} + 5\cdot 13^{5} + 10\cdot 13^{6} + 2\cdot 13^{7} +O(13^{8})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
$3$ | $2$ | $(1,3)$ | $1$ |
$3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
$4$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
$4$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
$4$ | $6$ | $(1,6,5,3,4,2)$ | $0$ |
$4$ | $6$ | $(1,2,4,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.