Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.49408.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.193.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.49408.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - 2x^{3} + x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 8 + \left(74 a + 34\right)\cdot 97 + \left(47 a + 22\right)\cdot 97^{2} + \left(83 a + 39\right)\cdot 97^{3} + \left(38 a + 10\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 a + 62 + \left(19 a + 3\right)\cdot 97 + \left(35 a + 55\right)\cdot 97^{2} + \left(55 a + 60\right)\cdot 97^{3} + \left(25 a + 29\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 93 + 66\cdot 97 + 79\cdot 97^{2} + 27\cdot 97^{3} + 71\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 35\cdot 97 + 67\cdot 97^{2} + 7\cdot 97^{3} + 20\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 a + 17 + \left(77 a + 68\right)\cdot 97 + \left(61 a + 70\right)\cdot 97^{2} + \left(41 a + 80\right)\cdot 97^{3} + \left(71 a + 96\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 71 a + 34 + \left(22 a + 82\right)\cdot 97 + \left(49 a + 92\right)\cdot 97^{2} + \left(13 a + 74\right)\cdot 97^{3} + \left(58 a + 62\right)\cdot 97^{4} +O(97^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.