Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(6875257828375\)\(\medspace = 5^{3} \cdot 3803^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.19015.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.19015.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.19015.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{3} - x^{2} - 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + 1 + \left(2 a + 2\right)\cdot 7 + a\cdot 7^{2} + a\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} +O(7^{5})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 6\cdot 7 + 2\cdot 7^{2} + 4\cdot 7^{3} + 3\cdot 7^{4} +O(7^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a + \left(6 a + 6\right)\cdot 7 + 4\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a + 2 + 3\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + \left(a + 6\right)\cdot 7^{3} + \left(a + 2\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a + 2 + \left(4 a + 3\right)\cdot 7 + \left(5 a + 6\right)\cdot 7^{2} + \left(5 a + 6\right)\cdot 7^{3} + a\cdot 7^{4} +O(7^{5})\)
| |
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.