Basic invariants
| Dimension: | $4$ |
| Group: | $F_5$ |
| Conductor: | \(392000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.392000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $F_5$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $F_5$ |
| Projective stem field: | Galois closure of 5.1.392000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 4x^{3} - 8x^{2} + 11x - 10 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 7\cdot 41 + 33\cdot 41^{2} + 25\cdot 41^{3} + 33\cdot 41^{4} + 37\cdot 41^{5} + 19\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 a + 9 + 7 a\cdot 41 + \left(20 a + 34\right)\cdot 41^{2} + \left(16 a + 26\right)\cdot 41^{3} + \left(36 a + 5\right)\cdot 41^{4} + \left(8 a + 23\right)\cdot 41^{5} + \left(27 a + 40\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 34 + \left(34 a + 10\right)\cdot 41 + \left(5 a + 13\right)\cdot 41^{2} + \left(34 a + 20\right)\cdot 41^{3} + \left(35 a + 17\right)\cdot 41^{4} + \left(19 a + 12\right)\cdot 41^{5} + \left(13 a + 5\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 17 + \left(33 a + 34\right)\cdot 41 + \left(20 a + 4\right)\cdot 41^{2} + \left(24 a + 15\right)\cdot 41^{3} + \left(4 a + 16\right)\cdot 41^{4} + \left(32 a + 13\right)\cdot 41^{5} + \left(13 a + 31\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 40 + \left(6 a + 28\right)\cdot 41 + \left(35 a + 37\right)\cdot 41^{2} + \left(6 a + 34\right)\cdot 41^{3} + \left(5 a + 8\right)\cdot 41^{4} + \left(21 a + 36\right)\cdot 41^{5} + \left(27 a + 25\right)\cdot 41^{6} +O(41^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $5$ | $4$ | $(1,4,2,3)$ | $0$ |
| $5$ | $4$ | $(1,3,2,4)$ | $0$ |
| $4$ | $5$ | $(1,2,3,5,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.