Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(270400\)\(\medspace = 2^{6} \cdot 5^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.270400.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.270400.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 5x^{3} - 2x^{2} + 7x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 16\cdot 31 + 8\cdot 31^{2} + 17\cdot 31^{3} + 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 25 + \left(17 a + 24\right)\cdot 31 + \left(10 a + 18\right)\cdot 31^{2} + \left(a + 11\right)\cdot 31^{3} + \left(11 a + 6\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 29 + \left(18 a + 6\right)\cdot 31 + \left(22 a + 8\right)\cdot 31^{2} + \left(18 a + 7\right)\cdot 31^{3} + \left(13 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 16 + \left(13 a + 17\right)\cdot 31 + \left(20 a + 22\right)\cdot 31^{2} + \left(29 a + 3\right)\cdot 31^{3} + \left(19 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 1 + \left(12 a + 27\right)\cdot 31 + \left(8 a + 3\right)\cdot 31^{2} + \left(12 a + 22\right)\cdot 31^{3} + \left(17 a + 17\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.