Normalized defining polynomial
\( x^{16} - 28x^{8} + 4 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $(4, 6)$ |
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| Discriminant: |
\(121029087867608368152576\)
\(\medspace = 2^{64}\cdot 3^{8}\)
|
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| Root discriminant: | \(27.71\) |
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| Galois root discriminant: | $2^{4}3^{1/2}\approx 27.712812921102035$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}+\frac{1}{4}$, $\frac{1}{16}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{3}{8}a$, $\frac{1}{16}a^{10}-\frac{1}{2}a^{6}-\frac{3}{8}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{2}a^{7}-\frac{3}{8}a^{3}$, $\frac{1}{16}a^{12}-\frac{3}{8}a^{4}$, $\frac{1}{16}a^{13}-\frac{3}{8}a^{5}$, $\frac{1}{16}a^{14}-\frac{3}{8}a^{6}$, $\frac{1}{16}a^{15}-\frac{3}{8}a^{7}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{1}{16}a^{14}+\frac{1}{16}a^{10}-\frac{15}{8}a^{6}-\frac{11}{8}a^{2}$, $\frac{1}{16}a^{12}-\frac{11}{8}a^{4}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{1}{16}a^{9}-\frac{15}{8}a^{5}+\frac{11}{8}a^{3}+\frac{11}{8}a$, $\frac{1}{8}a^{12}-\frac{15}{4}a^{4}+1$, $\frac{3}{16}a^{14}-\frac{41}{8}a^{6}+1$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{8}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{8}a^{8}-\frac{15}{8}a^{7}+\frac{15}{8}a^{6}+\frac{15}{8}a^{5}+\frac{11}{8}a^{4}-\frac{11}{4}a^{3}+\frac{11}{8}a^{2}+\frac{11}{8}a-\frac{3}{4}$, $\frac{1}{2}a^{15}+\frac{1}{4}a^{14}+\frac{1}{16}a^{13}-\frac{3}{16}a^{12}+\frac{3}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{8}a^{8}+14a^{7}-7a^{6}-\frac{15}{8}a^{5}+\frac{41}{8}a^{4}-\frac{41}{8}a^{3}+\frac{11}{8}a^{2}+\frac{11}{8}a-\frac{11}{4}$, $\frac{7}{16}a^{15}+\frac{1}{4}a^{13}-\frac{3}{16}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{97}{8}a^{7}-7a^{5}+\frac{41}{8}a^{4}+\frac{15}{4}a^{2}-\frac{11}{8}a+1$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}+\frac{15}{8}a^{5}+\frac{11}{8}a^{4}+\frac{19}{8}a^{3}-a^{2}-\frac{3}{8}a-1$
|
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| Regulator: | \( 203526.5980818538 \) |
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| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 203526.5980818538 \cdot 1}{2\cdot\sqrt{121029087867608368152576}}\cr\approx \mathstrut & 0.287968835714479 \end{aligned}\]
Galois group
$Q_{16}:C_2$ (as 16T32):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $Q_{16}:C_2$ |
| Character table for $Q_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt[4]{18})\), \(\Q(\sqrt[4]{2})\), \(\Q(\zeta_{24})^+\), 8.4.1358954496.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.1494186269970473680896.43 |
| 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 | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64g1.4565 | $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 16 x + 62$ | $16$ | $1$ | $64$ | 16T32 | $$[2, 3, 4, 5]^{2}$$ |
|
\(3\)
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |