Normalized defining polynomial
\( x^{8} - x^{7} + 2x^{6} + x^{5} - 15x^{4} + 23x^{3} + 6x^{2} - 27x + 19 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-8671400783\)
\(\medspace = -\,17^{4}\cdot 47^{3}\)
|
| |
| Root discriminant: | \(17.47\) |
| |
| Galois root discriminant: | $17^{1/2}47^{1/2}\approx 28.26658805020514$ | ||
| Ramified primes: |
\(17\), \(47\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{273}a^{7}-\frac{4}{273}a^{6}+\frac{2}{39}a^{5}-\frac{44}{91}a^{4}+\frac{17}{273}a^{3}-\frac{17}{39}a^{2}-\frac{92}{273}a-\frac{8}{91}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{12}{91}a^{7}-\frac{53}{273}a^{6}+\frac{7}{39}a^{5}-\frac{20}{273}a^{4}-\frac{571}{273}a^{3}+\frac{43}{13}a^{2}+\frac{328}{273}a-\frac{379}{91}$, $\frac{2}{273}a^{7}-\frac{8}{273}a^{6}+\frac{4}{39}a^{5}+\frac{3}{91}a^{4}+\frac{34}{273}a^{3}+\frac{5}{39}a^{2}-\frac{184}{273}a+\frac{75}{91}$, $\frac{16}{273}a^{7}+\frac{9}{91}a^{6}+\frac{2}{13}a^{5}+\frac{163}{273}a^{4}-\frac{92}{273}a^{3}+\frac{1}{39}a^{2}+\frac{530}{273}a-\frac{128}{91}$, $\frac{22}{273}a^{7}+\frac{1}{91}a^{6}+\frac{5}{39}a^{5}+\frac{33}{91}a^{4}-\frac{118}{91}a^{3}+\frac{29}{39}a^{2}+\frac{524}{273}a-\frac{346}{273}$
|
| |
| Regulator: | \( 81.706186763 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 81.706186763 \cdot 1}{2\cdot\sqrt{8671400783}}\cr\approx \mathstrut & 0.43529122614 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.13583.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.166101760110563345913601.2 |
| Degree 8 sibling: | 8.0.23973872753.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *16 | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.799.2t1.a.a | $1$ | $ 17 \cdot 47 $ | \(\Q(\sqrt{-799}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.799.4t3.a.a | $2$ | $ 17 \cdot 47 $ | 4.2.13583.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *16 | 2.799.8t6.a.a | $2$ | $ 17 \cdot 47 $ | 8.2.8671400783.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
| *16 | 2.799.8t6.a.b | $2$ | $ 17 \cdot 47 $ | 8.2.8671400783.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
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 *.