Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(163909850319\)\(\medspace = 3^{5} \cdot 877^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 5.3.23679.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | odd |
Projective image: | $S_5$ |
Projective field: | Galois closure of 5.3.23679.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 14 a + 14 + \left(13 a + 46\right)\cdot 47 + \left(9 a + 41\right)\cdot 47^{2} + \left(31 a + 30\right)\cdot 47^{3} + \left(5 a + 33\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 33 a + 42 + \left(33 a + 11\right)\cdot 47 + 37 a\cdot 47^{2} + \left(15 a + 37\right)\cdot 47^{3} + \left(41 a + 13\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 39 a + 11 + \left(22 a + 27\right)\cdot 47 + \left(45 a + 17\right)\cdot 47^{2} + \left(17 a + 7\right)\cdot 47^{3} + \left(24 a + 38\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 a + 42 + \left(24 a + 33\right)\cdot 47 + \left(a + 38\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 34 + 21\cdot 47 + 42\cdot 47^{2} + 20\cdot 47^{3} + 33\cdot 47^{4} +O(47^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |