Normalized defining polynomial
\( x^{8} + 16x^{6} - 4x^{4} + 256x^{2} - 44 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-3489136640000\)
\(\medspace = -\,2^{22}\cdot 5^{4}\cdot 11^{3}\)
|
| |
| Root discriminant: | \(36.97\) |
| |
| Galois root discriminant: | $2^{11/4}5^{1/2}11^{1/2}\approx 49.89003778087012$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{40}a^{4}-\frac{3}{10}a^{2}-\frac{7}{20}$, $\frac{1}{40}a^{5}-\frac{3}{10}a^{3}-\frac{7}{20}a$, $\frac{1}{120}a^{6}-\frac{1}{120}a^{4}-\frac{13}{60}a^{2}-\frac{17}{60}$, $\frac{1}{120}a^{7}-\frac{1}{120}a^{5}-\frac{13}{60}a^{3}-\frac{17}{60}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{120}a^{6}+\frac{17}{120}a^{4}-\frac{1}{60}a^{2}+\frac{37}{60}$, $\frac{1}{24}a^{6}+\frac{41}{60}a^{4}+\frac{13}{60}a^{2}+\frac{373}{30}$, $\frac{1}{20}a^{7}+\frac{37}{60}a^{6}+\frac{47}{20}a^{5}+\frac{1441}{120}a^{4}+\frac{249}{10}a^{3}+\frac{367}{15}a^{2}-\frac{113}{10}a-\frac{463}{60}$, $\frac{26881}{120}a^{7}-\frac{2231}{24}a^{6}+\frac{217381}{60}a^{5}-\frac{36055}{24}a^{4}-\frac{15931}{60}a^{3}+\frac{1571}{12}a^{2}+\frac{1719671}{30}a-\frac{285335}{12}$
|
| |
| Regulator: | \( 1913.6135089 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1913.6135089 \cdot 2}{2\cdot\sqrt{3489136640000}}\cr\approx \mathstrut & 1.0164706329 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.17600.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 sibling: | 8.0.7676100608000.2 |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(11\)
| 11.1.2.1a1.2 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
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 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.3520.4t3.d.a | $2$ | $ 2^{6} \cdot 5 \cdot 11 $ | 4.2.17600.2 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *16 | 2.14080.8t6.d.a | $2$ | $ 2^{8} \cdot 5 \cdot 11 $ | 8.2.3489136640000.8 | $D_{8}$ (as 8T6) | $1$ | $0$ |
| *16 | 2.14080.8t6.d.b | $2$ | $ 2^{8} \cdot 5 \cdot 11 $ | 8.2.3489136640000.8 | $D_{8}$ (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 *.