Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(557885504160000\)\(\medspace = 2^{8} \cdot 3^{20} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.118098000.1 |
| 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 6.2.118098000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 8x^{3} - 9x^{2} - 12x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 36 + 18\cdot 149 + 97\cdot 149^{2} + 144\cdot 149^{3} + 81\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 76 a + 41 + \left(91 a + 129\right)\cdot 149 + \left(85 a + 25\right)\cdot 149^{2} + \left(16 a + 99\right)\cdot 149^{3} + \left(39 a + 45\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 118 a + 71 + \left(142 a + 34\right)\cdot 149 + \left(126 a + 106\right)\cdot 149^{2} + \left(132 a + 115\right)\cdot 149^{3} + \left(18 a + 32\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 73 a + 47 + \left(57 a + 121\right)\cdot 149 + \left(63 a + 127\right)\cdot 149^{2} + \left(132 a + 79\right)\cdot 149^{3} + \left(109 a + 36\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 a + 96 + \left(6 a + 40\right)\cdot 149 + \left(22 a + 24\right)\cdot 149^{2} + \left(16 a + 73\right)\cdot 149^{3} + \left(130 a + 124\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 103\cdot 149 + 65\cdot 149^{2} + 83\cdot 149^{3} + 125\cdot 149^{4} +O(149^{5})\)
|
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.