Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(36100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.13032100.1 |
| 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 5.1.13032100.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 11x^{3} + 6x^{2} + 64x - 74 \)
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The roots of $f$ are computed in $\Q_{ 557 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O(557^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O(557^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O(557^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O(557^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O(557^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.