Normalized defining polynomial
\( x^{16} + 28x^{8} + 324 \)
Invariants
| Degree: | $16$ |
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| Signature: | $(0, 8)$ |
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| Discriminant: |
\(1494186269970473680896\)
\(\medspace = 2^{64}\cdot 3^{4}\)
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| Root discriminant: | \(21.06\) |
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| Galois root discriminant: | $2^{4}3^{1/2}\approx 27.712812921102035$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $Q_8$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\zeta_{8})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16}a^{8}-\frac{1}{2}a^{4}-\frac{1}{8}$, $\frac{1}{16}a^{9}-\frac{1}{2}a^{5}-\frac{1}{8}a$, $\frac{1}{48}a^{10}-\frac{1}{2}a^{6}-\frac{1}{24}a^{2}$, $\frac{1}{48}a^{11}-\frac{1}{2}a^{7}-\frac{1}{24}a^{3}$, $\frac{1}{144}a^{12}+\frac{23}{72}a^{4}$, $\frac{1}{288}a^{13}-\frac{1}{2}a^{7}-\frac{49}{144}a^{5}-\frac{1}{2}a$, $\frac{1}{864}a^{14}+\frac{95}{432}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{864}a^{15}+\frac{95}{432}a^{7}-\frac{1}{2}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( \frac{1}{72} a^{12} - \frac{1}{16} a^{8} + \frac{5}{36} a^{4} - \frac{7}{8} \)
(order $8$)
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| Fundamental units: |
$\frac{5}{864}a^{14}+\frac{1}{24}a^{10}+\frac{43}{432}a^{6}+\frac{5}{12}a^{2}$, $\frac{5}{864}a^{14}-\frac{1}{72}a^{12}+\frac{1}{16}a^{8}+\frac{43}{432}a^{6}-\frac{5}{36}a^{4}-\frac{1}{2}a^{2}+\frac{15}{8}$, $\frac{5}{864}a^{14}-\frac{1}{144}a^{12}+\frac{1}{24}a^{10}-\frac{1}{16}a^{8}-\frac{43}{432}a^{6}+\frac{13}{72}a^{4}+\frac{5}{12}a^{2}-\frac{7}{8}$, $\frac{7}{864}a^{15}+\frac{1}{72}a^{14}+\frac{1}{96}a^{13}+\frac{1}{144}a^{12}-\frac{1}{48}a^{10}-\frac{1}{16}a^{8}+\frac{17}{432}a^{7}+\frac{5}{36}a^{6}-\frac{1}{48}a^{5}-\frac{13}{72}a^{4}-\frac{1}{2}a^{3}-\frac{23}{24}a^{2}-\frac{1}{2}a-\frac{7}{8}$, $\frac{5}{432}a^{15}+\frac{5}{432}a^{14}-\frac{1}{96}a^{13}+\frac{1}{48}a^{12}-\frac{1}{48}a^{11}-\frac{43}{216}a^{7}+\frac{43}{216}a^{6}+\frac{1}{48}a^{5}-\frac{1}{24}a^{4}+\frac{1}{24}a^{3}-a^{2}+\frac{3}{2}a-1$, $\frac{1}{216}a^{15}-\frac{5}{216}a^{14}-\frac{1}{288}a^{13}+\frac{1}{18}a^{12}-\frac{1}{48}a^{11}-\frac{1}{24}a^{10}+\frac{1}{8}a^{9}-\frac{13}{108}a^{7}-\frac{43}{108}a^{6}+\frac{49}{144}a^{5}+\frac{5}{9}a^{4}-\frac{23}{24}a^{3}+\frac{1}{12}a^{2}+\frac{9}{4}a-2$, $\frac{1}{288}a^{15}-\frac{7}{432}a^{14}+\frac{1}{32}a^{13}-\frac{1}{144}a^{12}-\frac{1}{24}a^{11}+\frac{1}{24}a^{10}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{23}{144}a^{7}-\frac{17}{216}a^{6}+\frac{7}{16}a^{5}-\frac{23}{72}a^{4}-\frac{5}{12}a^{3}+\frac{11}{12}a^{2}+\frac{3}{8}a-\frac{11}{4}$
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| Regulator: | \( 39128.59405383013 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 39128.59405383013 \cdot 1}{8\cdot\sqrt{1494186269970473680896}}\cr\approx \mathstrut & 0.307355389243200 \end{aligned}\]
Galois group
$Q_{16}:C_2$ (as 16T50):
| 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
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt[4]{-2})\) x2, \(\Q(\sqrt[4]{2})\) x2, \(\Q(\zeta_{8})\), 8.0.16777216.2 |
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: | 16.4.121029087867608368152576.64 |
| Minimal sibling: | 16.4.121029087867608368152576.64 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64g1.589 | $x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 34$ | $16$ | $1$ | $64$ | 16T50 | $$[2, 3, 4, 5]^{2}$$ |
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |