Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(5058001\)\(\medspace = 13^{2} \cdot 173^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5058001.1 |
| 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 5.1.5058001.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 8x^{3} + x^{2} + 4x + 1 \)
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The roots of $f$ are computed in $\Q_{ 373 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 71 + 45\cdot 373 + 337\cdot 373^{2} + 59\cdot 373^{3} + 310\cdot 373^{4} +O(373^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 105 + 171\cdot 373 + 232\cdot 373^{2} + 184\cdot 373^{3} + 211\cdot 373^{4} +O(373^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 218 + 9\cdot 373 + 315\cdot 373^{2} + 81\cdot 373^{3} + 245\cdot 373^{4} +O(373^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 363 + 53\cdot 373 + 34\cdot 373^{2} + 53\cdot 373^{3} + 141\cdot 373^{4} +O(373^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 364 + 92\cdot 373 + 200\cdot 373^{2} + 366\cdot 373^{3} + 210\cdot 373^{4} +O(373^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |