Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(13222479725\)\(\medspace = 5^{2} \cdot 7^{2} \cdot 13^{3} \cdot 17^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.9475375.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.221.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.9475375.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 6x^{4} - 6x^{3} + 9x^{2} - 10x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 47 a + 26 + \left(14 a + 42\right)\cdot 71 + \left(26 a + 3\right)\cdot 71^{2} + \left(68 a + 33\right)\cdot 71^{3} + \left(7 a + 43\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 27 + \left(41 a + 23\right)\cdot 71 + \left(58 a + 23\right)\cdot 71^{2} + 10\cdot 71^{3} + \left(34 a + 21\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 70 a + 29 + \left(29 a + 33\right)\cdot 71 + \left(12 a + 28\right)\cdot 71^{2} + \left(70 a + 24\right)\cdot 71^{3} + \left(36 a + 17\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 49 + \left(56 a + 24\right)\cdot 71 + \left(44 a + 41\right)\cdot 71^{2} + \left(2 a + 1\right)\cdot 71^{3} + \left(63 a + 62\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 + 46\cdot 71 + 20\cdot 71^{2} + 70\cdot 71^{3} + 6\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 + 42\cdot 71 + 24\cdot 71^{2} + 2\cdot 71^{3} + 62\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(2,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,6,5)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.