Normalized defining polynomial
\( x^{16} - 72x^{12} + 324x^{8} + 11664x^{4} + 26244 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[4, 6]$ |
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| Discriminant: |
\(7745861623526935561764864\)
\(\medspace = 2^{70}\cdot 3^{8}\)
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| Root discriminant: | \(35.94\) |
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| Galois root discriminant: | $2^{305/64}3^{1/2}\approx 47.11472932657837$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{648}a^{8}-\frac{1}{54}a^{6}+\frac{1}{4}$, $\frac{1}{648}a^{9}-\frac{1}{54}a^{7}+\frac{1}{4}a$, $\frac{1}{1944}a^{10}+\frac{1}{12}a^{2}$, $\frac{1}{1944}a^{11}+\frac{1}{12}a^{3}$, $\frac{1}{5832}a^{12}+\frac{1}{36}a^{4}$, $\frac{1}{5832}a^{13}+\frac{1}{36}a^{5}$, $\frac{1}{34992}a^{14}-\frac{1}{54}a^{7}+\frac{1}{216}a^{6}-\frac{1}{18}a^{4}$, $\frac{1}{34992}a^{15}+\frac{1}{216}a^{7}-\frac{1}{18}a^{5}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{1}{1944}a^{10}-\frac{1}{27}a^{6}+\frac{5}{12}a^{2}$, $\frac{1}{1944}a^{10}-\frac{1}{27}a^{6}+\frac{1}{12}a^{2}$, $\frac{1}{8748}a^{14}+\frac{1}{5832}a^{12}+\frac{17}{1944}a^{10}-\frac{1}{72}a^{8}-\frac{2}{27}a^{6}+\frac{5}{36}a^{4}-\frac{5}{4}a^{2}+\frac{7}{4}$, $\frac{1}{17496}a^{14}+\frac{1}{216}a^{10}-\frac{5}{108}a^{6}-\frac{7}{12}a^{2}$, $\frac{1}{8748}a^{14}-\frac{17}{1944}a^{10}-\frac{1}{648}a^{8}+\frac{2}{27}a^{6}+\frac{1}{9}a^{4}+\frac{5}{4}a^{2}-\frac{1}{4}$, $\frac{5}{17496}a^{15}-\frac{1}{17496}a^{14}+\frac{1}{2916}a^{13}+\frac{1}{1458}a^{12}+\frac{41}{1944}a^{11}+\frac{7}{1944}a^{10}-\frac{17}{648}a^{9}-\frac{17}{324}a^{8}-\frac{5}{36}a^{7}-\frac{1}{108}a^{6}+\frac{1}{6}a^{5}+\frac{1}{3}a^{4}-\frac{35}{12}a^{3}-\frac{5}{12}a^{2}+\frac{15}{4}a+\frac{15}{2}$, $\frac{1}{17496}a^{15}-\frac{1}{4374}a^{14}+\frac{1}{2916}a^{13}-\frac{1}{5832}a^{12}-\frac{1}{216}a^{11}+\frac{35}{1944}a^{10}-\frac{17}{648}a^{9}+\frac{5}{324}a^{8}+\frac{7}{108}a^{7}-\frac{5}{27}a^{6}+\frac{5}{18}a^{5}-\frac{5}{36}a^{4}+\frac{1}{4}a^{3}-\frac{13}{12}a^{2}+\frac{7}{4}a-\frac{3}{2}$, $\frac{13}{8748}a^{15}+\frac{1}{324}a^{14}-\frac{23}{2916}a^{13}+\frac{2}{243}a^{12}-\frac{1}{9}a^{11}-\frac{56}{243}a^{10}+\frac{191}{324}a^{9}-\frac{11}{18}a^{8}+\frac{43}{54}a^{7}+\frac{29}{18}a^{6}-\frac{25}{6}a^{5}+4a^{4}+\frac{44}{3}a^{3}+32a^{2}-\frac{161}{2}a+90$, $\frac{31}{34992}a^{15}-\frac{5}{4374}a^{14}+\frac{13}{5832}a^{13}-\frac{23}{5832}a^{12}-\frac{113}{1944}a^{11}+\frac{133}{1944}a^{10}-\frac{83}{648}a^{9}+\frac{5}{24}a^{8}+\frac{11}{216}a^{7}+\frac{5}{18}a^{6}-\frac{11}{12}a^{5}+\frac{101}{36}a^{4}+\frac{31}{12}a^{3}+\frac{7}{4}a^{2}-\frac{7}{4}a+\frac{11}{4}$
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| Regulator: | \( 2364275.9773017 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 2364275.9773017 \cdot 1}{2\cdot\sqrt{7745861623526935561764864}}\cr\approx \mathstrut & 0.41815038360334 \end{aligned}\] (assuming GRH)
Galois group
$C_4^2.(C_2\times C_4)$ (as 16T362):
| A solvable group of order 128 |
| The 14 conjugacy class representatives for $C_4^2.(C_2\times C_4)$ |
| Character table for $C_4^2.(C_2\times C_4)$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.268435456.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 32 siblings: | deg 32, deg 32, deg 32, deg 32, deg 32 |
| Arithmetically equivalent sibling: | 16.4.7745861623526935561764864.184 |
| Minimal sibling: | 16.4.7745861623526935561764864.184 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.70e1.3401 | $x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 16 x^{7} + 24 x^{4} + 2$ | $16$ | $1$ | $70$ | 16T362 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]$$ |
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |