Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(341\)\(\medspace = 11 \cdot 31 \) |
| Artin field: | Galois closure of 5.5.13521270961.2 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{341}(190,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 136x^{3} - 41x^{2} + 3039x - 1431 \)
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The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 28\cdot 47 + 9\cdot 47^{2} + 46\cdot 47^{3} + 34\cdot 47^{4} +O(47^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 15 + 13\cdot 47 + 7\cdot 47^{2} + 47^{3} + 2\cdot 47^{4} +O(47^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 17 + 13\cdot 47 + 19\cdot 47^{2} + 15\cdot 47^{3} + 31\cdot 47^{4} +O(47^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 27 + 14\cdot 47 + 44\cdot 47^{2} + 19\cdot 47^{3} + 36\cdot 47^{4} +O(47^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 33 + 24\cdot 47 + 13\cdot 47^{2} + 11\cdot 47^{3} + 36\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,3,5,4,2)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,2,3,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,2,4,5,3)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.