Normalized defining polynomial
\( x^{16} + 252x^{8} + 26244 \)
Invariants
| Degree: | $16$ |
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| Signature: | $(0, 8)$ |
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| Discriminant: |
\(9803356117276277820358656\)
\(\medspace = 2^{64}\cdot 3^{12}\)
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| Root discriminant: | \(36.47\) |
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| Galois root discriminant: | $2^{4}3^{3/4}\approx 36.47211291127644$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{144}a^{8}-\frac{1}{6}a^{4}-\frac{1}{8}$, $\frac{1}{144}a^{9}-\frac{1}{6}a^{5}-\frac{1}{8}a$, $\frac{1}{432}a^{10}-\frac{1}{6}a^{6}-\frac{1}{24}a^{2}$, $\frac{1}{1296}a^{11}-\frac{1}{6}a^{7}+\frac{23}{72}a^{3}$, $\frac{1}{3888}a^{12}+\frac{23}{216}a^{4}$, $\frac{1}{7776}a^{13}-\frac{1}{6}a^{7}-\frac{49}{432}a^{5}-\frac{1}{2}a$, $\frac{1}{23328}a^{14}+\frac{95}{1296}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{69984}a^{15}+\frac{95}{3888}a^{7}-\frac{1}{2}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( \frac{1}{1944} a^{12} - \frac{1}{144} a^{8} + \frac{5}{108} a^{4} - \frac{7}{8} \)
(order $8$)
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| Fundamental units: |
$\frac{1}{1296}a^{12}-\frac{1}{144}a^{8}+\frac{11}{72}a^{4}+\frac{1}{8}$, $\frac{7}{23328}a^{14}+\frac{1}{3888}a^{12}+\frac{1}{432}a^{10}+\frac{1}{72}a^{8}-\frac{17}{1296}a^{6}+\frac{23}{216}a^{4}+\frac{11}{24}a^{2}+\frac{11}{4}$, $\frac{1}{7776}a^{14}+\frac{1}{1296}a^{12}-\frac{1}{432}a^{10}+\frac{23}{432}a^{6}-\frac{1}{72}a^{4}-\frac{11}{24}a^{2}+1$, $\frac{29}{34992}a^{15}+\frac{37}{23328}a^{14}-\frac{11}{3888}a^{13}+\frac{19}{3888}a^{12}-\frac{1}{162}a^{11}+\frac{1}{432}a^{10}+\frac{1}{72}a^{9}-\frac{1}{18}a^{8}-\frac{163}{1944}a^{7}+\frac{275}{1296}a^{6}-\frac{109}{216}a^{5}+\frac{221}{216}a^{4}-\frac{14}{9}a^{3}+\frac{35}{24}a^{2}+\frac{3}{4}a-8$, $\frac{1}{2916}a^{15}+\frac{7}{11664}a^{14}+\frac{1}{3888}a^{13}+\frac{1}{1944}a^{12}+\frac{7}{1296}a^{11}-\frac{1}{72}a^{10}-\frac{1}{72}a^{9}+\frac{1}{18}a^{8}-\frac{7}{81}a^{7}+\frac{17}{648}a^{6}+\frac{23}{216}a^{5}-\frac{13}{108}a^{4}+\frac{89}{72}a^{3}-\frac{3}{4}a^{2}-\frac{19}{4}a+6$, $\frac{7}{17496}a^{15}-\frac{5}{23328}a^{14}+\frac{5}{3888}a^{13}-\frac{13}{3888}a^{12}+\frac{1}{648}a^{11}+\frac{1}{72}a^{9}-\frac{1}{24}a^{8}-\frac{17}{972}a^{7}-\frac{43}{1296}a^{6}+\frac{43}{216}a^{5}-\frac{155}{216}a^{4}+\frac{23}{36}a^{3}-\frac{1}{2}a^{2}+\frac{3}{4}a-\frac{17}{4}$, $\frac{7}{17496}a^{15}-\frac{1}{7776}a^{14}+\frac{5}{3888}a^{13}+\frac{11}{3888}a^{12}-\frac{1}{648}a^{11}+\frac{1}{432}a^{10}+\frac{1}{72}a^{9}-\frac{1}{36}a^{8}+\frac{17}{972}a^{7}-\frac{23}{432}a^{6}+\frac{43}{216}a^{5}+\frac{37}{216}a^{4}-\frac{23}{36}a^{3}+\frac{11}{24}a^{2}+\frac{3}{4}a-\frac{17}{2}$
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| Regulator: | \( 3603057.554814803 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3603057.554814803 \cdot 2}{8\cdot\sqrt{9803356117276277820358656}}\cr\approx \mathstrut & 0.698815880193543 \end{aligned}\]
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
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt[4]{-18})\) x2, \(\Q(\sqrt[4]{18})\) x2, \(\Q(\zeta_{8})\), 8.0.1358954496.9 |
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: | 16.4.9803356117276277820358656.271 |
| Minimal sibling: | 16.4.9803356117276277820358656.271 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64g1.644 | $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 50$ | $16$ | $1$ | $64$ | 16T50 | $$[2, 3, 4, 5]^{2}$$ |
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |