Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(1865461525\)\(\medspace = 5^{2} \cdot 421^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.373092305.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T34 |
Parity: | even |
Determinant: | 1.421.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.2.373092305.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + x^{4} - 21x^{3} + 21x^{2} + 5 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 7 a + 5 + \left(25 a + 5\right)\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(16 a + 8\right)\cdot 31^{3} + \left(27 a + 24\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 24 a + 19 + \left(5 a + 17\right)\cdot 31 + \left(25 a + 23\right)\cdot 31^{2} + \left(14 a + 3\right)\cdot 31^{3} + \left(3 a + 1\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 8 + 8\cdot 31 + 31^{2} + 19\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 16 a + 13 + \left(15 a + 28\right)\cdot 31 + \left(18 a + 15\right)\cdot 31^{2} + \left(15 a + 6\right)\cdot 31^{3} + \left(28 a + 13\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 5 + 21\cdot 31 + 8\cdot 31^{2} + 5\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 15 a + 14 + \left(15 a + 12\right)\cdot 31 + \left(12 a + 6\right)\cdot 31^{2} + \left(15 a + 19\right)\cdot 31^{3} + \left(2 a + 23\right)\cdot 31^{4} +O(31^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$6$ | $2$ | $(1,2)$ | $0$ |
$9$ | $2$ | $(1,2)(4,5)$ | $0$ |
$4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$4$ | $3$ | $(1,2,3)$ | $-2$ |
$18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
$12$ | $6$ | $(1,5,2,6,3,4)$ | $1$ |
$12$ | $6$ | $(1,2)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.