Normalized defining polynomial
\( x^{5} - x^{4} - 28x^{3} - 37x^{2} + 25x - 1 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $[5, 0]$ |
| |
| Discriminant: |
\(25411681\)
\(\medspace = 71^{4}\)
|
| |
| Root discriminant: | \(30.27\) |
| |
| Galois root discriminant: | $71^{4/5}\approx 30.269599790850265$ | ||
| Ramified primes: |
\(71\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_5$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(71\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{71}(1,·)$, $\chi_{71}(25,·)$, $\chi_{71}(5,·)$, $\chi_{71}(54,·)$, $\chi_{71}(57,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{23}a^{4}+\frac{7}{23}a^{3}+\frac{5}{23}a^{2}+\frac{3}{23}a+\frac{3}{23}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{23}a^{4}-\frac{9}{23}a^{3}-\frac{36}{23}a^{2}+\frac{75}{23}a+\frac{52}{23}$, $a$, $\frac{6}{23}a^{4}-\frac{4}{23}a^{3}-\frac{177}{23}a^{2}-\frac{258}{23}a+\frac{156}{23}$, $\frac{1}{23}a^{4}+\frac{7}{23}a^{3}-\frac{64}{23}a^{2}-\frac{135}{23}a+\frac{72}{23}$
|
| |
| Regulator: | \( 70.6106756434 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{0}\cdot 70.6106756434 \cdot 1}{2\cdot\sqrt{25411681}}\cr\approx \mathstrut & 0.224116407517 \end{aligned}\]
Galois group
| A cyclic group of order 5 |
| The 5 conjugacy class representatives for $C_5$ |
| Character table for $C_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.1.0.1}{1} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.1.0.1}{1} }^{5}$ | ${\href{/padicField/41.1.0.1}{1} }^{5}$ | ${\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(71\)
| 71.1.5.4a1.1 | $x^{5} + 71$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.71.5t1.a.a | $1$ | $ 71 $ | 5.5.25411681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.71.5t1.a.b | $1$ | $ 71 $ | 5.5.25411681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.71.5t1.a.c | $1$ | $ 71 $ | 5.5.25411681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.71.5t1.a.d | $1$ | $ 71 $ | 5.5.25411681.1 | $C_5$ (as 5T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.