Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.26873856.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.2.26873856.5 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{3} + 9x^{2} - 6 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 107 a + 64 + \left(98 a + 25\right)\cdot 113 + \left(37 a + 54\right)\cdot 113^{2} + \left(55 a + 87\right)\cdot 113^{3} + \left(63 a + 38\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 49\cdot 113 + 74\cdot 113^{2} + 76\cdot 113^{3} + 99\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 105 + \left(14 a + 88\right)\cdot 113 + \left(75 a + 70\right)\cdot 113^{2} + \left(57 a + 35\right)\cdot 113^{3} + \left(49 a + 67\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 18 + \left(49 a + 26\right)\cdot 113 + \left(54 a + 32\right)\cdot 113^{2} + \left(44 a + 103\right)\cdot 113^{3} + \left(62 a + 75\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 109 + 108\cdot 113 + 35\cdot 113^{2} + 18\cdot 113^{3} + 68\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 103 a + 25 + \left(63 a + 40\right)\cdot 113 + \left(58 a + 71\right)\cdot 113^{2} + \left(68 a + 17\right)\cdot 113^{3} + \left(50 a + 102\right)\cdot 113^{4} +O(113^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.