Basic invariants
Dimension: | $1$ |
Group: | $C_5$ |
Conductor: | \(71\) |
Artin field: | Galois closure of 5.5.25411681.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_5$ |
Parity: | even |
Dirichlet character: | \(\chi_{71}(25,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 28x^{3} - 37x^{2} + 25x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 4 + 4\cdot 37 + 30\cdot 37^{2} + 6\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})\)
$r_{ 2 }$ |
$=$ |
\( 22 + 36\cdot 37 + 12\cdot 37^{2} + 33\cdot 37^{3} + 22\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 23 + 18\cdot 37 + 10\cdot 37^{2} + 11\cdot 37^{3} + 13\cdot 37^{4} +O(37^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 + 34\cdot 37 + 32\cdot 37^{2} + 2\cdot 37^{3} + 31\cdot 37^{4} +O(37^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 36 + 16\cdot 37 + 24\cdot 37^{2} + 19\cdot 37^{3} + 37^{4} +O(37^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $5$ | $(1,5,2,4,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,4,5,3,2)$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,3,4,2,5)$ | $\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.