Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.25000000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.25000000.3 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 10 + \left(24 a + 21\right)\cdot 53 + \left(24 a + 20\right)\cdot 53^{2} + \left(8 a + 51\right)\cdot 53^{3} + \left(41 a + 30\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 39 + \left(29 a + 19\right)\cdot 53 + \left(12 a + 24\right)\cdot 53^{2} + \left(21 a + 8\right)\cdot 53^{3} + \left(39 a + 36\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 25 + \left(23 a + 8\right)\cdot 53 + \left(40 a + 45\right)\cdot 53^{2} + \left(31 a + 27\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 + 50\cdot 53 + 27\cdot 53^{2} + 10\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 34 + \left(28 a + 5\right)\cdot 53 + \left(28 a + 41\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(11 a + 28\right)\cdot 53^{4} +O(53^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |