Normalized defining polynomial
\( x^{8} - 41x^{6} + 533x^{4} - 2296x^{2} + 656 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[8, 0]$ |
| |
| Discriminant: |
\(49857094113536\)
\(\medspace = 2^{8}\cdot 41^{7}\)
|
| |
| Root discriminant: | \(51.55\) |
| |
| Galois root discriminant: | $2\cdot 41^{7/8}\approx 51.548480764667524$ | ||
| Ramified primes: |
\(2\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_8$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(164=2^{2}\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(3,·)$, $\chi_{164}(73,·)$, $\chi_{164}(55,·)$, $\chi_{164}(79,·)$, $\chi_{164}(81,·)$, $\chi_{164}(9,·)$, $\chi_{164}(27,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{368}a^{6}+\frac{5}{16}a^{4}+\frac{73}{368}a^{2}-\frac{27}{92}$, $\frac{1}{736}a^{7}+\frac{5}{32}a^{5}+\frac{73}{736}a^{3}+\frac{65}{184}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{184}a^{6}-\frac{7}{8}a^{4}+\frac{1285}{184}a^{2}-\frac{365}{46}$, $\frac{5}{368}a^{6}-\frac{7}{16}a^{4}+\frac{1101}{368}a^{2}+\frac{49}{92}$, $\frac{25}{368}a^{6}-\frac{35}{16}a^{4}+\frac{5873}{368}a^{2}-\frac{399}{92}$, $\frac{33}{736}a^{7}-\frac{29}{184}a^{6}-\frac{43}{32}a^{5}+\frac{39}{8}a^{4}+\frac{6457}{736}a^{3}-\frac{6349}{184}a^{2}+\frac{765}{184}a+\frac{323}{46}$, $\frac{9}{368}a^{7}-\frac{43}{368}a^{6}-\frac{11}{16}a^{5}+\frac{57}{16}a^{4}+\frac{1209}{368}a^{3}-\frac{9395}{368}a^{2}+\frac{1091}{92}a+\frac{1345}{92}$, $\frac{5}{736}a^{7}+\frac{29}{184}a^{6}-\frac{7}{32}a^{5}-\frac{39}{8}a^{4}+\frac{1101}{736}a^{3}+\frac{6349}{184}a^{2}+\frac{325}{184}a-\frac{415}{46}$, $\frac{17}{184}a^{7}-\frac{101}{368}a^{6}-\frac{23}{8}a^{5}+\frac{135}{16}a^{4}+\frac{3909}{184}a^{3}-\frac{22093}{368}a^{2}-\frac{333}{23}a+\frac{2083}{92}$
|
| |
| Regulator: | \( 27148.8915251 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 27148.8915251 \cdot 1}{2\cdot\sqrt{49857094113536}}\cr\approx \mathstrut & 0.492151248018 \end{aligned}\]
Galois group
| A cyclic group of order 8 |
| The 8 conjugacy class representatives for $C_8$ |
| Character table for $C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a1.2 | $x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$ | $2$ | $2$ | $4$ | $C_4$ | $$[2]^{2}$$ |
| 2.2.2.4a1.2 | $x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$ | $2$ | $2$ | $4$ | $C_4$ | $$[2]^{2}$$ | |
|
\(41\)
| 41.1.8.7a1.1 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $$[\ ]_{8}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *8 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *8 | 1.164.8t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | 8.8.49857094113536.1 | $C_8$ (as 8T1) | $0$ | $1$ |
| *8 | 1.41.4t1.a.a | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| *8 | 1.164.8t1.a.b | $1$ | $ 2^{2} \cdot 41 $ | 8.8.49857094113536.1 | $C_8$ (as 8T1) | $0$ | $1$ |
| *8 | 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.164.8t1.a.c | $1$ | $ 2^{2} \cdot 41 $ | 8.8.49857094113536.1 | $C_8$ (as 8T1) | $0$ | $1$ |
| *8 | 1.41.4t1.a.b | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| *8 | 1.164.8t1.a.d | $1$ | $ 2^{2} \cdot 41 $ | 8.8.49857094113536.1 | $C_8$ (as 8T1) | $0$ | $1$ |
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 *.