Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(845\)\(\medspace = 5 \cdot 13^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.142805.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | even |
Determinant: | 1.5.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.4225.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + x^{3} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 28 a + 16 + \left(17 a + 24\right)\cdot 31 + \left(17 a + 3\right)\cdot 31^{2} + \left(13 a + 11\right)\cdot 31^{3} + \left(a + 22\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 9 a + 27 + \left(15 a + 25\right)\cdot 31 + \left(11 a + 14\right)\cdot 31^{2} + 25\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 17 + 7\cdot 31 + 19\cdot 31^{2} + 28\cdot 31^{3} + 29\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 11 + 17\cdot 31 + 11\cdot 31^{2} + 23\cdot 31^{3} + 26\cdot 31^{4} +O(31^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 a + 14 + \left(15 a + 16\right)\cdot 31 + \left(19 a + 22\right)\cdot 31^{2} + \left(30 a + 14\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 6 }$ | $=$ | \( 3 a + 10 + \left(13 a + 1\right)\cdot 31 + \left(13 a + 21\right)\cdot 31^{2} + \left(17 a + 20\right)\cdot 31^{3} + \left(29 a + 11\right)\cdot 31^{4} +O(31^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
$3$ | $2$ | $(1,6)$ | $1$ |
$3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
$4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
$4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
$4$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
$4$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.