Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(45508\)\(\medspace = 2^{2} \cdot 31 \cdot 367 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.45508.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.0.45508.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$:
\( x^{2} + 306x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 27 a + 292 + \left(103 a + 278\right)\cdot 307 + \left(126 a + 206\right)\cdot 307^{2} + \left(117 a + 255\right)\cdot 307^{3} + \left(33 a + 190\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 280 a + 12 + \left(203 a + 48\right)\cdot 307 + \left(180 a + 230\right)\cdot 307^{2} + \left(189 a + 246\right)\cdot 307^{3} + \left(273 a + 106\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 186 a + 305 + \left(59 a + 58\right)\cdot 307 + \left(282 a + 47\right)\cdot 307^{2} + \left(209 a + 67\right)\cdot 307^{3} + \left(251 a + 17\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 + 121\cdot 307 + 49\cdot 307^{2} + 180\cdot 307^{3} + 274\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 121 a + 184 + \left(247 a + 239\right)\cdot 307 + \left(24 a + 269\right)\cdot 307^{2} + \left(97 a + 301\right)\cdot 307^{3} + \left(55 a + 58\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 79 + 174\cdot 307 + 117\cdot 307^{2} + 176\cdot 307^{3} + 272\cdot 307^{4} +O(307^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |