Normalized defining polynomial
\( x^{8} - 2x^{7} + 3x^{6} - 4x^{4} + 2x^{3} + 4x^{2} - 4x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(20123648\)
\(\medspace = 2^{12}\cdot 17^{3}\)
|
| |
| Root discriminant: | \(8.18\) |
| |
| Galois root discriminant: | $2^{3/2}17^{3/4}\approx 23.679999262753405$ | ||
| Ramified primes: |
\(2\), \(17\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$
| Monogenic: | Not computed | |
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{3}a^{7}-3a^{6}+4a^{5}+a^{4}-\frac{20}{3}a^{3}+a^{2}+\frac{23}{3}a-\frac{10}{3}$, $a^{7}-a^{6}+2a^{5}+2a^{4}-2a^{3}+4a-1$, $a$
|
| |
| Regulator: | \( 1.81338098584 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.81338098584 \cdot 1}{2\cdot\sqrt{20123648}}\cr\approx \mathstrut & 0.315010608308 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.1088.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.12a1.1 | $x^{8} + 2 x^{5} + 2 x^{4} + x^{2} + 2 x + 3$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $$[3]^{4}$$ |
|
\(17\)
| 17.4.1.0a1.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 17.1.4.3a1.2 | $x^{4} + 51$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.136.2t1.b.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{-34}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.136.4t1.b.a | $1$ | $ 2^{3} \cdot 17 $ | 4.0.314432.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.136.4t1.b.b | $1$ | $ 2^{3} \cdot 17 $ | 4.0.314432.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 2.2312.4t3.a.a | $2$ | $ 2^{3} \cdot 17^{2}$ | 4.0.314432.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 2.136.4t3.d.a | $2$ | $ 2^{3} \cdot 17 $ | 4.2.2312.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *32 | 2.136.8t17.a.a | $2$ | $ 2^{3} \cdot 17 $ | 8.0.20123648.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
| 2.2312.8t17.a.a | $2$ | $ 2^{3} \cdot 17^{2}$ | 8.0.20123648.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
| *32 | 2.136.8t17.a.b | $2$ | $ 2^{3} \cdot 17 $ | 8.0.20123648.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
| 2.2312.8t17.a.b | $2$ | $ 2^{3} \cdot 17^{2}$ | 8.0.20123648.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
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 *.