Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(23225\)\(\medspace = 5^{2} \cdot 929 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.4008825445.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:D_4$ |
Parity: | even |
Determinant: | 1.929.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.2.4008825445.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - x^{4} + 35x^{3} - 32x^{2} - 33x + 40 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 2\cdot 11 + 11^{2} + 7\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\)
$r_{ 2 }$ |
$=$ |
\( a + 1 + \left(2 a + 7\right)\cdot 11 + \left(a + 8\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + \left(5 a + 8\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 a + 7 + \left(6 a + 6\right)\cdot 11 + \left(9 a + 5\right)\cdot 11^{2} + \left(3 a + 4\right)\cdot 11^{3} + \left(4 a + 10\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 10 a + 5 + \left(8 a + 3\right)\cdot 11 + 9 a\cdot 11^{2} + \left(8 a + 1\right)\cdot 11^{3} + \left(5 a + 5\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 5 }$ |
$=$ |
\( a + 3 + \left(4 a + 2\right)\cdot 11 + \left(a + 4\right)\cdot 11^{2} + \left(7 a + 10\right)\cdot 11^{3} + \left(6 a + 1\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 6 + 2\cdot 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
$6$ | $2$ | $(1,3)$ | $0$ |
$9$ | $2$ | $(1,3)(2,4)$ | $0$ |
$4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
$4$ | $3$ | $(2,4,6)$ | $-2$ |
$18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
$12$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$12$ | $6$ | $(1,3)(2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.