Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(107584\)\(\medspace = 2^{6} \cdot 41^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.180848704.1 |
| 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 5.1.180848704.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 16x^{3} + 36x^{2} + 267x - 319 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 20\cdot 67 + 55\cdot 67^{2} + 61\cdot 67^{3} + 46\cdot 67^{4} + 56\cdot 67^{5} +O(67^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 28 + 53\cdot 67 + 47\cdot 67^{2} + 14\cdot 67^{3} + 9\cdot 67^{4} + 67^{5} +O(67^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 49 + 62\cdot 67 + 53\cdot 67^{2} + 7\cdot 67^{3} + 11\cdot 67^{4} + 55\cdot 67^{5} +O(67^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 56 + 21\cdot 67 + 25\cdot 67^{2} + 32\cdot 67^{3} + 11\cdot 67^{4} + 22\cdot 67^{5} +O(67^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 66 + 42\cdot 67 + 18\cdot 67^{2} + 17\cdot 67^{3} + 55\cdot 67^{4} + 65\cdot 67^{5} +O(67^{6})\)
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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 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.