Normalized defining polynomial
\( x^{5} - x^{4} - 268x^{3} - 1020x^{2} + 4128x - 1024 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $[5, 0]$ |
| |
| Discriminant: |
\(202716958081\)
\(\medspace = 11^{4}\cdot 61^{4}\)
|
| |
| Root discriminant: | \(182.55\) |
| |
| Galois root discriminant: | $11^{4/5}61^{4/5}\approx 182.5483938364549$ | ||
| Ramified primes: |
\(11\), \(61\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_5$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(671=11\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{671}(400,·)$, $\chi_{671}(1,·)$, $\chi_{671}(619,·)$, $\chi_{671}(20,·)$, $\chi_{671}(302,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{3424}a^{4}-\frac{85}{3424}a^{3}-\frac{26}{107}a^{2}-\frac{117}{856}a+\frac{20}{107}$
| Monogenic: | No | |
| Index: | $32$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ |
| |
| Narrow class group: | $C_{5}$, which has order $5$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1167}{428}a^{4}+\frac{373}{107}a^{3}-\frac{309149}{428}a^{2}-\frac{945359}{214}a+\frac{131829}{107}$, $\frac{607}{856}a^{4}-\frac{6655}{856}a^{3}-\frac{20647}{214}a^{2}+\frac{14581}{214}a-\frac{1195}{107}$, $\frac{181}{1712}a^{4}+\frac{879}{1712}a^{3}-\frac{1597}{214}a^{2}+\frac{6215}{428}a-\frac{357}{107}$, $\frac{2989}{856}a^{4}-\frac{1973}{856}a^{3}-\frac{201095}{214}a^{2}-\frac{822225}{214}a+\frac{1442335}{107}$
|
| |
| Regulator: | \( 25011.6652454 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{0}\cdot 25011.6652454 \cdot 5}{2\cdot\sqrt{202716958081}}\cr\approx \mathstrut & 4.44413818296 \end{aligned}\]
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 | ${\href{/padicField/2.1.0.1}{1} }^{5}$ | ${\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.1.5.4a1.1 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
|
\(61\)
| 61.1.5.4a1.2 | $x^{5} + 122$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.671.5t1.a.a | $1$ | $ 11 \cdot 61 $ | 5.5.202716958081.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.671.5t1.a.b | $1$ | $ 11 \cdot 61 $ | 5.5.202716958081.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.671.5t1.a.c | $1$ | $ 11 \cdot 61 $ | 5.5.202716958081.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.671.5t1.a.d | $1$ | $ 11 \cdot 61 $ | 5.5.202716958081.1 | $C_5$ (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 *.