Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(15043\)\(\medspace = 7^{2} \cdot 307 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.737107.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.307.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.4618201.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - x^{4} + 8x^{3} - 7x^{2} - 7x + 7 \) . |
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:
$r_{ 1 }$ | $=$ | \( 39 + 19\cdot 41 + 35\cdot 41^{2} + 5\cdot 41^{3} + 32\cdot 41^{4} + 9\cdot 41^{5} +O(41^{6})\) |
$r_{ 2 }$ | $=$ | \( 2 a + 16 + \left(40 a + 36\right)\cdot 41 + \left(6 a + 16\right)\cdot 41^{2} + \left(9 a + 36\right)\cdot 41^{3} + \left(27 a + 9\right)\cdot 41^{4} + \left(27 a + 26\right)\cdot 41^{5} +O(41^{6})\) |
$r_{ 3 }$ | $=$ | \( 33 + 4\cdot 41 + 35\cdot 41^{2} + 37\cdot 41^{3} + 9\cdot 41^{4} + 3\cdot 41^{5} +O(41^{6})\) |
$r_{ 4 }$ | $=$ | \( 11 a + 32 + \left(30 a + 36\right)\cdot 41 + \left(9 a + 39\right)\cdot 41^{2} + \left(36 a + 4\right)\cdot 41^{3} + \left(14 a + 31\right)\cdot 41^{4} + \left(a + 26\right)\cdot 41^{5} +O(41^{6})\) |
$r_{ 5 }$ | $=$ | \( 30 a + 24 + \left(10 a + 34\right)\cdot 41 + \left(31 a + 38\right)\cdot 41^{2} + \left(4 a + 21\right)\cdot 41^{3} + \left(26 a + 39\right)\cdot 41^{4} + \left(39 a + 15\right)\cdot 41^{5} +O(41^{6})\) |
$r_{ 6 }$ | $=$ | \( 39 a + 22 + 31\cdot 41 + \left(34 a + 38\right)\cdot 41^{2} + \left(31 a + 15\right)\cdot 41^{3} + 13 a\cdot 41^{4} + 13 a\cdot 41^{5} +O(41^{6})\) |
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 value |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
$3$ | $2$ | $(1,3)$ | $1$ |
$3$ | $2$ | $(1,3)(2,6)$ | $-1$ |
$4$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
$4$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
$4$ | $6$ | $(1,6,5,3,2,4)$ | $0$ |
$4$ | $6$ | $(1,4,2,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.