Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(154927809\)\(\medspace = 3^{6} \cdot 461^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.12447.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.12447.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + x^{3} - x^{2} - 3x - 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 17 + 30\cdot 47 + 32\cdot 47^{2} + 11\cdot 47^{3} + 39\cdot 47^{4} +O(47^{5})\)
$r_{ 2 }$ |
$=$ |
\( 38 a + 22 + \left(24 a + 32\right)\cdot 47 + \left(35 a + 17\right)\cdot 47^{2} + \left(29 a + 35\right)\cdot 47^{3} + \left(3 a + 6\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a + 4 + \left(22 a + 44\right)\cdot 47 + \left(11 a + 16\right)\cdot 47^{2} + \left(17 a + 12\right)\cdot 47^{3} + \left(43 a + 31\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 13 a + 36 + \left(7 a + 39\right)\cdot 47 + \left(17 a + 46\right)\cdot 47^{2} + \left(43 a + 5\right)\cdot 47^{3} + \left(13 a + 16\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 34 a + 15 + \left(39 a + 41\right)\cdot 47 + \left(29 a + 26\right)\cdot 47^{2} + \left(3 a + 28\right)\cdot 47^{3} + 33 a\cdot 47^{4} +O(47^{5})\)
| |
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 value |
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $-1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.