Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(341\)\(\medspace = 11 \cdot 31 \) |
| Artin field: | Galois closure of 5.5.13521270961.4 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{341}(159,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 136x^{3} + 641x^{2} - 371x - 67 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13\cdot 67 + 61\cdot 67^{2} + 54\cdot 67^{3} + 26\cdot 67^{4} +O(67^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 1 + 44\cdot 67 + 59\cdot 67^{2} + 11\cdot 67^{3} + 43\cdot 67^{4} +O(67^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 25 + 43\cdot 67 + 27\cdot 67^{2} + 31\cdot 67^{3} + 15\cdot 67^{4} +O(67^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 53 + 48\cdot 67 + 37\cdot 67^{2} + 30\cdot 67^{3} + 18\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 51\cdot 67 + 14\cdot 67^{2} + 5\cdot 67^{3} + 30\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $5$ | $(1,3,5,4,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | |
| $1$ | $5$ | $(1,5,2,3,4)$ | $\zeta_{5}^{3}$ | |
| $1$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}^{2}$ | |
| $1$ | $5$ | $(1,2,4,5,3)$ | $\zeta_{5}$ |