Normalized defining polynomial
\( x^{43} - 5 \)
Invariants
| Degree: | $43$ |
| |
| Signature: | $(1, 21)$ |
| |
| Discriminant: |
\(-394\!\cdots\!875\)
\(\medspace = -\,5^{42}\cdot 43^{43}\)
|
| |
| Root discriminant: | \(207.10\) |
| |
| Galois root discriminant: | $5^{42/43}43^{1847/1806}\approx 225.5623000662761$ | ||
| Ramified primes: |
\(5\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
|
Unit group
| Rank: | $21$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: | not computed |
| |
| Regulator: | not computed |
| |
| Unit signature rank: | not computed |
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 = \mathstrut &\frac{2^{1}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{3943517496516112150835347458453732578688700991509015683067847748815326440308126620948314666748046875}}\cr\mathstrut & \text{
Galois group
| A solvable group of order 1806 |
| The 43 conjugacy class representatives for $F_{43}$ |
| Character table for $F_{43}$ |
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 | ${\href{/padicField/2.14.0.1}{14} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $42{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $42{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $42{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | R | ${\href{/padicField/47.7.0.1}{7} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $43$ | $43$ | $1$ | $42$ | |||
|
\(43\)
| Deg $43$ | $43$ | $1$ | $43$ |