Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(45223\)\(\medspace = 41 \cdot 1103 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.45223.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.0.45223.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 18\cdot 19 + 4\cdot 19^{2} + 17\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 8 a\cdot 19 + \left(8 a + 16\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(14 a + 7\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 2 + \left(10 a + 6\right)\cdot 19 + \left(10 a + 16\right)\cdot 19^{2} + \left(13 a + 5\right)\cdot 19^{3} + \left(4 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 17 + \left(16 a + 8\right)\cdot 19 + \left(14 a + 4\right)\cdot 19^{2} + \left(9 a + 11\right)\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 19 + 12\cdot 19^{2} + 7\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 1 + \left(2 a + 3\right)\cdot 19 + \left(4 a + 3\right)\cdot 19^{2} + \left(9 a + 6\right)\cdot 19^{3} + \left(18 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |