Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(8746082375\)\(\medspace = 5^{3} \cdot 19^{3} \cdot 101^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.86594875.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.86594875.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 6x^{4} + 5x^{3} + 20x^{2} + 25x + 25 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 13 + 10\cdot 13^{2} + 5\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 7 + 7 a\cdot 13 + 10 a\cdot 13^{2} + \left(6 a + 12\right)\cdot 13^{3} + \left(3 a + 9\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 4 + \left(5 a + 11\right)\cdot 13 + \left(2 a + 2\right)\cdot 13^{2} + \left(6 a + 8\right)\cdot 13^{3} + \left(9 a + 6\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 12\cdot 13 + 3\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 9\cdot 13 + 4\cdot 13^{2} + \left(2 a + 12\right)\cdot 13^{3} + \left(11 a + 2\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 5 + \left(12 a + 4\right)\cdot 13 + \left(12 a + 4\right)\cdot 13^{2} + \left(10 a + 1\right)\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} +O(13^{5})\)
|
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.