Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(7102225\)\(\medspace = 5^{2} \cdot 13^{2} \cdot 41^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.2665.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.2665.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + x^{3} - x^{2} + 2x - 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 2 + 21\cdot 23 + 4\cdot 23^{2} + 23^{3} + 5\cdot 23^{4} +O(23^{5})\)
$r_{ 2 }$ |
$=$ |
\( 10 + 12\cdot 23 + 15\cdot 23^{2} + 5\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 9 + 9\cdot 23 + 2\cdot 23^{2} + 11\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 13 a + \left(3 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(17 a + 16\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a + 3 + \left(19 a + 10\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(5 a + 11\right)\cdot 23^{3} + \left(13 a + 6\right)\cdot 23^{4} +O(23^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $-1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.