Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(5653\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5653.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.5653.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.5653.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 2x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 20 + \left(10 a + 10\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(18 a + 17\right)\cdot 23^{3} + 8 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 2\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 16 + \left(12 a + 11\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + 14 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 9 + \left(21 a + 17\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(19 a + 21\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 8 + \left(a + 3\right)\cdot 23 + \left(17 a + 5\right)\cdot 23^{2} + \left(3 a + 8\right)\cdot 23^{3} + \left(5 a + 8\right)\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |