Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(2617\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.2617.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.2617.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.2617.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + x^{3} - 2x^{2} - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 11 a + 13 + \left(16 a + 17\right)\cdot 29 + \left(12 a + 18\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(4 a + 19\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 18 a + 10 + \left(12 a + 1\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(7 a + 7\right)\cdot 29^{3} + \left(24 a + 22\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 20 a + 20 + \left(10 a + 21\right)\cdot 29 + \left(27 a + 26\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(14 a + 27\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 9 a + 4 + \left(18 a + 26\right)\cdot 29 + \left(a + 7\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(14 a + 1\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 11 + 20\cdot 29 + 26\cdot 29^{2} + 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
| |
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.