Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(39204\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.4743684.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.4743684.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 2 + \left(22 a + 29\right)\cdot 37 + \left(36 a + 15\right)\cdot 37^{2} + 27\cdot 37^{3} + \left(4 a + 14\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 32\cdot 37 + 32\cdot 37^{2} + 24\cdot 37^{3} + 8\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 32 + \left(14 a + 19\right)\cdot 37 + 28\cdot 37^{2} + \left(36 a + 31\right)\cdot 37^{3} + \left(32 a + 29\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + \left(6 a + 8\right)\cdot 37 + \left(17 a + 4\right)\cdot 37^{2} + 29 a\cdot 37^{3} + \left(33 a + 13\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 11 + \left(30 a + 21\right)\cdot 37 + \left(19 a + 29\right)\cdot 37^{2} + \left(7 a + 26\right)\cdot 37^{3} + \left(3 a + 7\right)\cdot 37^{4} +O(37^{5})\)
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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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |