Basic invariants
| Dimension: | $5$ |
| Group: | $A_5$ |
| Conductor: | \(15405625\)\(\medspace = 5^{4} \cdot 157^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.616225.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PSL(2,5)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.616225.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 5x^{3} + 6x^{2} + 13x - 1 \)
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The roots of $f$ are computed in $\Q_{ 421 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 70 + 261\cdot 421 + 276\cdot 421^{2} + 162\cdot 421^{3} + 340\cdot 421^{4} +O(421^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 247 + 163\cdot 421 + 319\cdot 421^{2} + 234\cdot 421^{3} + 220\cdot 421^{4} +O(421^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 305 + 12\cdot 421 + 124\cdot 421^{2} + 78\cdot 421^{3} + 343\cdot 421^{4} +O(421^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 320 + 102\cdot 421 + 21\cdot 421^{2} + 198\cdot 421^{3} + 278\cdot 421^{4} +O(421^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 322 + 301\cdot 421 + 100\cdot 421^{2} + 168\cdot 421^{3} + 80\cdot 421^{4} +O(421^{5})\)
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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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.