Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(1777\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.1777.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1777.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.1777.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + x^{3} - 2x^{2} + x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 a + 8 + \left(11 a + 8\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(17 a + 13\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 20 a + 14 + \left(11 a + 4\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a + 3 + \left(16 a + 17\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(8 a + 3\right)\cdot 23^{3} + \left(18 a + 7\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( 15 + 14\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 21 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O(23^{5})\) |
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 |
$1$ | $1$ | $()$ | $4$ | |
$10$ | $2$ | $(1,2)$ | $2$ | |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
$20$ | $3$ | $(1,2,3)$ | $1$ | |
$30$ | $4$ | $(1,2,3,4)$ | $0$ | |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |