Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(202500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.5062500.1 |
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 5.1.5062500.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 10x^{2} - 24 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 20 + 17\cdot 31 + 3\cdot 31^{2} + 30\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 25 a + \left(9 a + 17\right)\cdot 31 + \left(4 a + 27\right)\cdot 31^{2} + \left(2 a + 25\right)\cdot 31^{3} + \left(12 a + 5\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 6 a + 19 + \left(21 a + 11\right)\cdot 31 + \left(26 a + 26\right)\cdot 31^{2} + \left(28 a + 25\right)\cdot 31^{3} + \left(18 a + 27\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 11 a + 16 + \left(26 a + 2\right)\cdot 31 + \left(20 a + 10\right)\cdot 31^{2} + \left(28 a + 18\right)\cdot 31^{3} + \left(9 a + 11\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 20 a + 7 + \left(4 a + 13\right)\cdot 31 + \left(10 a + 25\right)\cdot 31^{2} + \left(2 a + 23\right)\cdot 31^{3} + \left(21 a + 2\right)\cdot 31^{4} +O(31^{5})\)
| |
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.