Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(149769\)\(\medspace = 3^{4} \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.149769.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.149769.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 19 + \left(22 a + 21\right)\cdot 29 + \left(14 a + 2\right)\cdot 29^{2} + \left(3 a + 1\right)\cdot 29^{3} + \left(6 a + 26\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 5 + \left(14 a + 18\right)\cdot 29 + 21 a\cdot 29^{2} + \left(5 a + 21\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 9 + \left(14 a + 10\right)\cdot 29 + \left(7 a + 6\right)\cdot 29^{2} + \left(23 a + 28\right)\cdot 29^{3} + \left(16 a + 13\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 12 + \left(6 a + 2\right)\cdot 29 + \left(14 a + 25\right)\cdot 29^{2} + \left(25 a + 3\right)\cdot 29^{3} + \left(22 a + 24\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 5\cdot 29 + 23\cdot 29^{2} + 3\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |