Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(4169\)\(\medspace = 11 \cdot 379 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.4169.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.4169.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.4169.1 |
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:
$r_{ 1 }$ | $=$ | \( a + 4 + 3 a\cdot 7 + \left(a + 3\right)\cdot 7^{2} + 5\cdot 7^{3} +O(7^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 5 + \left(3 a + 2\right)\cdot 7 + \left(5 a + 1\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + 6 a\cdot 7^{4} +O(7^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 + 7^{2} + 7^{3} +O(7^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 a + \left(5 a + 5\right)\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} +O(7^{5})\) |
$r_{ 5 }$ | $=$ | \( a + 6 + \left(a + 4\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(4 a + 1\right)\cdot 7^{3} + \left(3 a + 3\right)\cdot 7^{4} +O(7^{5})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $4$ | |
$10$ | $2$ | $(1,2)$ | $2$ | |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
$20$ | $3$ | $(1,2,3)$ | $1$ | |
$30$ | $4$ | $(1,2,3,4)$ | $0$ | |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |