Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(102090816\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 421^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.102090816.1 |
| 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 5.5.102090816.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 18x^{3} - 14x^{2} + 51x + 30 \)
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The roots of $f$ are computed in $\Q_{ 521 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 42 + 73\cdot 521 + 274\cdot 521^{2} + 447\cdot 521^{3} + 509\cdot 521^{4} +O(521^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 84 + 96\cdot 521 + 332\cdot 521^{2} + 290\cdot 521^{3} + 309\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 220 + 73\cdot 521 + 189\cdot 521^{2} + 182\cdot 521^{3} + 503\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 277 + 101\cdot 521 + 458\cdot 521^{2} + 276\cdot 521^{3} + 151\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 419 + 176\cdot 521 + 309\cdot 521^{2} + 365\cdot 521^{3} + 88\cdot 521^{4} +O(521^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.