Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(14202376626313\)\(\medspace = 61^{3} \cdot 397^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.24217.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.24217.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.24217.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 5 + 19\cdot 47 + 14\cdot 47^{2} + 31\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\)
$r_{ 2 }$ |
$=$ |
\( 22 a + 6 + \left(9 a + 26\right)\cdot 47 + \left(3 a + 10\right)\cdot 47^{2} + \left(21 a + 28\right)\cdot 47^{3} + \left(31 a + 18\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 25 a + 15 + \left(13 a + 35\right)\cdot 47 + \left(46 a + 14\right)\cdot 47^{2} + \left(24 a + 5\right)\cdot 47^{3} + \left(43 a + 20\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 22 a + 18 + \left(33 a + 37\right)\cdot 47 + 46\cdot 47^{2} + \left(22 a + 8\right)\cdot 47^{3} + \left(3 a + 35\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 25 a + 3 + \left(37 a + 23\right)\cdot 47 + \left(43 a + 7\right)\cdot 47^{2} + \left(25 a + 20\right)\cdot 47^{3} + \left(15 a + 13\right)\cdot 47^{4} +O(47^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $-2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.