Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(618252462359\)\(\medspace = 7^{3} \cdot 1217^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.8519.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.8519.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.8519.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 6 a + 13 + \left(10 a + 17\right)\cdot 43 + 27\cdot 43^{2} + \left(26 a + 2\right)\cdot 43^{3} + \left(2 a + 6\right)\cdot 43^{4} +O(43^{5})\)
$r_{ 2 }$ |
$=$ |
\( 30 a + 5 + \left(9 a + 10\right)\cdot 43 + \left(33 a + 12\right)\cdot 43^{2} + \left(22 a + 32\right)\cdot 43^{3} + \left(24 a + 11\right)\cdot 43^{4} +O(43^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 15 + 4\cdot 43 + 36\cdot 43^{2} + 29\cdot 43^{4} +O(43^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 13 a + 35 + \left(33 a + 32\right)\cdot 43 + \left(9 a + 35\right)\cdot 43^{2} + \left(20 a + 21\right)\cdot 43^{3} + \left(18 a + 13\right)\cdot 43^{4} +O(43^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 37 a + 19 + \left(32 a + 21\right)\cdot 43 + \left(42 a + 17\right)\cdot 43^{2} + \left(16 a + 28\right)\cdot 43^{3} + \left(40 a + 25\right)\cdot 43^{4} +O(43^{5})\)
| |
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.