Basic invariants
| Dimension: | $4$ |
| Group: | $F_5$ |
| Conductor: | \(2178000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.2178000.1 |
| 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 5.1.2178000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + 10x^{3} + 20x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 10\cdot 29 + 25\cdot 29^{2} + 7\cdot 29^{3} + 19\cdot 29^{4} + 19\cdot 29^{5} + 13\cdot 29^{6} + 25\cdot 29^{7} + 17\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 13 + \left(a + 19\right)\cdot 29 + \left(7 a + 2\right)\cdot 29^{2} + \left(21 a + 18\right)\cdot 29^{3} + \left(8 a + 12\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} + \left(7 a + 14\right)\cdot 29^{6} + \left(4 a + 24\right)\cdot 29^{7} + \left(a + 28\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 10 + \left(27 a + 15\right)\cdot 29 + \left(21 a + 7\right)\cdot 29^{2} + \left(7 a + 1\right)\cdot 29^{3} + \left(20 a + 6\right)\cdot 29^{4} + \left(26 a + 9\right)\cdot 29^{5} + \left(21 a + 18\right)\cdot 29^{6} + \left(24 a + 9\right)\cdot 29^{7} + \left(27 a + 1\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 8 + \left(21 a + 6\right)\cdot 29 + 20\cdot 29^{2} + \left(20 a + 23\right)\cdot 29^{3} + \left(5 a + 5\right)\cdot 29^{4} + \left(12 a + 27\right)\cdot 29^{5} + \left(2 a + 5\right)\cdot 29^{6} + \left(23 a + 15\right)\cdot 29^{7} + \left(3 a + 21\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 26 + \left(7 a + 6\right)\cdot 29 + \left(28 a + 2\right)\cdot 29^{2} + \left(8 a + 7\right)\cdot 29^{3} + \left(23 a + 14\right)\cdot 29^{4} + \left(16 a + 24\right)\cdot 29^{5} + \left(26 a + 5\right)\cdot 29^{6} + \left(5 a + 12\right)\cdot 29^{7} + \left(25 a + 17\right)\cdot 29^{8} +O(29^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,4)(3,5)$ | $0$ |
| $5$ | $4$ | $(1,5,4,3)$ | $0$ |
| $5$ | $4$ | $(1,3,4,5)$ | $0$ |
| $4$ | $5$ | $(1,5,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.