Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(186\!\cdots\!401\)\(\medspace = 35099^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.35099.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.35099.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} + x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 57\cdot 59 + 10\cdot 59^{2} + 51\cdot 59^{3} + 47\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 46 + \left(53 a + 47\right)\cdot 59 + \left(26 a + 22\right)\cdot 59^{2} + \left(23 a + 51\right)\cdot 59^{3} + \left(4 a + 30\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 + 14\cdot 59 + 50\cdot 59^{2} + 14\cdot 59^{3} + 42\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 a + 58 + \left(5 a + 29\right)\cdot 59 + \left(32 a + 55\right)\cdot 59^{2} + \left(35 a + 47\right)\cdot 59^{3} + \left(54 a + 11\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 34\cdot 59 + 25\cdot 59^{2} + 42\cdot 59^{3} + 6\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 51\cdot 59 + 11\cdot 59^{2} + 28\cdot 59^{3} + 37\cdot 59^{4} +O(59^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.