Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(50625\)\(\medspace = 3^{4} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.31640625.2 |
| 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 5.1.31640625.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 25x^{2} - 75 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 13 + 4\cdot 13^{2} + 8\cdot 13^{4} + 9\cdot 13^{5} + 6\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + \left(3 a + 5\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(9 a + 12\right)\cdot 13^{3} + \left(a + 7\right)\cdot 13^{4} + \left(9 a + 2\right)\cdot 13^{5} + \left(7 a + 11\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 3 + \left(9 a + 5\right)\cdot 13 + \left(9 a + 6\right)\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + 11 a\cdot 13^{4} + \left(3 a + 10\right)\cdot 13^{5} + \left(5 a + 9\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 1 + \left(10 a + 10\right)\cdot 13 + \left(3 a + 7\right)\cdot 13^{2} + \left(7 a + 8\right)\cdot 13^{3} + \left(7 a + 4\right)\cdot 13^{4} + 10 a\cdot 13^{5} + 8 a\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 7 + \left(2 a + 1\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(5 a + 12\right)\cdot 13^{3} + \left(5 a + 4\right)\cdot 13^{4} + \left(2 a + 3\right)\cdot 13^{5} + \left(4 a + 11\right)\cdot 13^{6} +O(13^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.