Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(23104\)\(\medspace = 2^{6} \cdot 19^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.8340544.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.4.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.23104.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 2x^{4} - 5x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 22 a + 30 + \left(3 a + 3\right)\cdot 37 + \left(2 a + 16\right)\cdot 37^{2} + \left(17 a + 22\right)\cdot 37^{3} + \left(20 a + 4\right)\cdot 37^{4} + \left(21 a + 4\right)\cdot 37^{5} + \left(29 a + 7\right)\cdot 37^{6} + \left(19 a + 12\right)\cdot 37^{7} + \left(a + 25\right)\cdot 37^{8} +O(37^{9})\)
$r_{ 2 }$ |
$=$ |
\( 14 a + 9 + \left(10 a + 23\right)\cdot 37 + \left(10 a + 21\right)\cdot 37^{2} + \left(11 a + 19\right)\cdot 37^{3} + \left(21 a + 18\right)\cdot 37^{4} + \left(10 a + 26\right)\cdot 37^{5} + \left(25 a + 28\right)\cdot 37^{6} + \left(20 a + 26\right)\cdot 37^{7} + \left(21 a + 22\right)\cdot 37^{8} +O(37^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 9 + 5\cdot 37 + 36\cdot 37^{2} + 35\cdot 37^{3} + 20\cdot 37^{4} + 23\cdot 37^{5} + 27\cdot 37^{6} + 25\cdot 37^{7} + 34\cdot 37^{8} +O(37^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 15 a + 7 + \left(33 a + 33\right)\cdot 37 + \left(34 a + 20\right)\cdot 37^{2} + \left(19 a + 14\right)\cdot 37^{3} + \left(16 a + 32\right)\cdot 37^{4} + \left(15 a + 32\right)\cdot 37^{5} + \left(7 a + 29\right)\cdot 37^{6} + \left(17 a + 24\right)\cdot 37^{7} + \left(35 a + 11\right)\cdot 37^{8} +O(37^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 23 a + 28 + \left(26 a + 13\right)\cdot 37 + \left(26 a + 15\right)\cdot 37^{2} + \left(25 a + 17\right)\cdot 37^{3} + \left(15 a + 18\right)\cdot 37^{4} + \left(26 a + 10\right)\cdot 37^{5} + \left(11 a + 8\right)\cdot 37^{6} + \left(16 a + 10\right)\cdot 37^{7} + \left(15 a + 14\right)\cdot 37^{8} +O(37^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 28 + 31\cdot 37 + 37^{3} + 16\cdot 37^{4} + 13\cdot 37^{5} + 9\cdot 37^{6} + 11\cdot 37^{7} + 2\cdot 37^{8} +O(37^{9})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
$3$ | $2$ | $(1,4)$ | $1$ |
$3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
$4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
$4$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
$4$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.