Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(181057\)\(\medspace = 331 \cdot 547 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.181057.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.181057.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.181057.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} - 4x^{3} + 7x^{2} + 2x - 3 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 35 a + 6 + \left(35 a + 9\right)\cdot 37 + \left(23 a + 15\right)\cdot 37^{2} + \left(20 a + 11\right)\cdot 37^{3} + a\cdot 37^{4} +O(37^{5})\)
$r_{ 2 }$ |
$=$ |
\( 22 a + 15 + \left(5 a + 1\right)\cdot 37 + \left(27 a + 22\right)\cdot 37^{2} + \left(17 a + 24\right)\cdot 37^{3} + \left(35 a + 31\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a + 35 + \left(a + 6\right)\cdot 37 + \left(13 a + 1\right)\cdot 37^{2} + \left(16 a + 33\right)\cdot 37^{3} + \left(35 a + 22\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 15 a + 29 + \left(31 a + 1\right)\cdot 37 + \left(9 a + 14\right)\cdot 37^{2} + \left(19 a + 31\right)\cdot 37^{3} + \left(a + 7\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 28 + 17\cdot 37 + 21\cdot 37^{2} + 10\cdot 37^{3} + 11\cdot 37^{4} +O(37^{5})\)
| |
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.