Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(5477930481596041\)\(\medspace = 7^{3} \cdot 25183^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.176281.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | even |
Determinant: | 1.176281.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.176281.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 5x^{3} + 3x^{2} + 4x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 10 a + 4 + 12\cdot 13 + \left(a + 7\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( a + 9 + \left(5 a + 7\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + \left(11 a + 5\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 3 + 4\cdot 13 + 3\cdot 13^{2} + 2\cdot 13^{3} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 a + 1 + \left(12 a + 3\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(12 a + 4\right)\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 12 a + 10 + \left(7 a + 11\right)\cdot 13 + \left(11 a + 7\right)\cdot 13^{2} + \left(5 a + 9\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O(13^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.