Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(161307172017\)\(\medspace = 3^{2} \cdot 2617^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.70659.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.2617.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.70659.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 4x^{2} + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 65\cdot 67 + 17\cdot 67^{2} + 10\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 43 a + 62 + \left(24 a + 11\right)\cdot 67 + \left(25 a + 64\right)\cdot 67^{2} + \left(5 a + 29\right)\cdot 67^{3} + \left(36 a + 26\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 56 + 34\cdot 67 + 38\cdot 67^{2} + 10\cdot 67^{3} + 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 33 + 42 a\cdot 67 + \left(41 a + 7\right)\cdot 67^{2} + \left(61 a + 26\right)\cdot 67^{3} + \left(30 a + 31\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 28 + \left(33 a + 52\right)\cdot 67 + \left(35 a + 15\right)\cdot 67^{2} + \left(35 a + 42\right)\cdot 67^{3} + \left(59 a + 44\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 52 a + 21 + \left(33 a + 36\right)\cdot 67 + \left(31 a + 57\right)\cdot 67^{2} + \left(31 a + 14\right)\cdot 67^{3} + \left(7 a + 46\right)\cdot 67^{4} +O(67^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ | ✓ |
| $6$ | $2$ | $(2,4)$ | $-2$ | |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ | |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ | |
| $4$ | $3$ | $(1,2,4)$ | $1$ | |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ | |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $0$ | |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $1$ |