Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(2680810943141213\)\(\medspace = 138917^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.138917.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.138917.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.138917.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 6x^{3} - 2x^{2} + 3x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{2} + 60x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 55 a + 5 + \left(38 a + 32\right)\cdot 61 + \left(22 a + 40\right)\cdot 61^{2} + \left(58 a + 48\right)\cdot 61^{3} + \left(54 a + 42\right)\cdot 61^{4} +O(61^{5})\)
$r_{ 2 }$ |
$=$ |
\( 26 + 29\cdot 61 + 10\cdot 61^{2} + 45\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 34 a + 29 + \left(18 a + 60\right)\cdot 61 + \left(50 a + 37\right)\cdot 61^{2} + \left(56 a + 29\right)\cdot 61^{3} + \left(57 a + 17\right)\cdot 61^{4} +O(61^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a + 60 + \left(22 a + 15\right)\cdot 61 + \left(38 a + 24\right)\cdot 61^{2} + \left(2 a + 23\right)\cdot 61^{3} + \left(6 a + 39\right)\cdot 61^{4} +O(61^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 27 a + 2 + \left(42 a + 45\right)\cdot 61 + \left(10 a + 8\right)\cdot 61^{2} + \left(4 a + 36\right)\cdot 61^{3} + \left(3 a + 18\right)\cdot 61^{4} +O(61^{5})\)
| |
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.