Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(45301\)\(\medspace = 89 \cdot 509 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.45301.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.45301.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.45301.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$:
\( x^{2} + 274x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 8 + \left(3 a + 205\right)\cdot 277 + \left(145 a + 227\right)\cdot 277^{2} + \left(170 a + 240\right)\cdot 277^{3} + \left(139 a + 186\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 232 a + 121 + \left(208 a + 203\right)\cdot 277 + \left(147 a + 104\right)\cdot 277^{2} + \left(240 a + 32\right)\cdot 277^{3} + \left(73 a + 182\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 275 a + 14 + \left(273 a + 212\right)\cdot 277 + \left(131 a + 105\right)\cdot 277^{2} + \left(106 a + 53\right)\cdot 277^{3} + \left(137 a + 158\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a + 263 + \left(68 a + 43\right)\cdot 277 + \left(129 a + 62\right)\cdot 277^{2} + \left(36 a + 52\right)\cdot 277^{3} + \left(203 a + 163\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 90 a + 78 + \left(86 a + 137\right)\cdot 277 + \left(107 a + 47\right)\cdot 277^{2} + \left(161 a + 176\right)\cdot 277^{3} + \left(244 a + 199\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 187 a + 71 + \left(190 a + 29\right)\cdot 277 + \left(169 a + 6\right)\cdot 277^{2} + \left(115 a + 276\right)\cdot 277^{3} + \left(32 a + 217\right)\cdot 277^{4} +O(277^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.