Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(33620\)\(\medspace = 2^{2} \cdot 5 \cdot 41^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.1378420.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:D_4$ |
Parity: | even |
Determinant: | 1.5.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.2.1378420.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 4x^{4} - x^{3} + 8x^{2} + 6x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 3 a + 6 + \left(22 a + 19\right)\cdot 31 + \left(6 a + 14\right)\cdot 31^{2} + \left(22 a + 25\right)\cdot 31^{3} + \left(2 a + 21\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 a + \left(24 a + 24\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(2 a + 5\right)\cdot 31^{3} + \left(6 a + 7\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 18 a + 26 + \left(6 a + 28\right)\cdot 31 + \left(12 a + 3\right)\cdot 31^{2} + \left(28 a + 23\right)\cdot 31^{3} + \left(24 a + 16\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 + 4\cdot 31 + 26\cdot 31^{2} + 25\cdot 31^{3} + 12\cdot 31^{4} +O(31^{5})\) |
$r_{ 5 }$ | $=$ | \( 14 + 18\cdot 31 + 20\cdot 31^{2} + 11\cdot 31^{3} + 29\cdot 31^{4} +O(31^{5})\) |
$r_{ 6 }$ | $=$ | \( 28 a + 12 + \left(8 a + 29\right)\cdot 31 + \left(24 a + 5\right)\cdot 31^{2} + \left(8 a + 1\right)\cdot 31^{3} + \left(28 a + 5\right)\cdot 31^{4} +O(31^{5})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$6$ | $2$ | $(2,3)$ | $2$ |
$9$ | $2$ | $(1,4)(2,3)$ | $0$ |
$4$ | $3$ | $(2,3,5)$ | $1$ |
$4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
$18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
$12$ | $6$ | $(1,2,4,3,6,5)$ | $0$ |
$12$ | $6$ | $(1,4,6)(2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.