Normalized defining polynomial
\( x^{16} + 8x^{14} + 4x^{12} - 72x^{10} - 80x^{8} + 64x^{6} + 120x^{4} + 48x^{2} + 4 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[4, 6]$ |
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| Discriminant: |
\(30257271966902092038144\)
\(\medspace = 2^{62}\cdot 3^{8}\)
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| Root discriminant: | \(25.41\) |
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| Galois root discriminant: | $2^{137/32}3^{1/2}\approx 33.677922740001875$ | ||
| Ramified primes: |
\(2\), \(3\)
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{5}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{12}+\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{13}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{14}+\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{15}+\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$
| 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: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{5}{8}a^{14}-\frac{17}{4}a^{12}+\frac{11}{4}a^{10}+\frac{171}{4}a^{8}-\frac{7}{4}a^{6}-48a^{4}-18a^{2}+\frac{1}{2}$, $9a^{15}-\frac{517}{8}a^{13}+17a^{11}+\frac{5075}{8}a^{9}+200a^{7}-743a^{5}-471a^{3}-\frac{169}{4}a$, $\frac{11}{8}a^{14}+\frac{39}{4}a^{12}-\frac{27}{8}a^{10}-96a^{8}-\frac{45}{2}a^{6}+\frac{219}{2}a^{4}+\frac{253}{4}a^{2}+6$, $\frac{5}{2}a^{14}-\frac{145}{8}a^{12}+\frac{29}{8}a^{10}+\frac{355}{2}a^{8}+\frac{267}{4}a^{6}-\frac{849}{4}a^{4}-\frac{561}{4}a^{2}-\frac{23}{2}$, $\frac{75}{8}a^{15}+\frac{539}{8}a^{13}-\frac{69}{4}a^{11}-\frac{5285}{8}a^{9}-\frac{851}{4}a^{7}+770a^{5}+498a^{3}+\frac{187}{4}a$, $\frac{33}{8}a^{14}-\frac{237}{8}a^{12}+\frac{31}{4}a^{10}+\frac{1163}{4}a^{8}+\frac{369}{4}a^{6}-\frac{1359}{4}a^{4}-218a^{2}-22$, $\frac{17}{8}a^{14}+\frac{125}{8}a^{12}-\frac{13}{8}a^{10}-152a^{8}-\frac{143}{2}a^{6}+\frac{729}{4}a^{4}+\frac{547}{4}a^{2}+\frac{29}{2}$, $\frac{5}{4}a^{15}-\frac{17}{8}a^{14}+9a^{13}-\frac{123}{8}a^{12}-\frac{17}{8}a^{11}+\frac{13}{4}a^{10}-\frac{351}{4}a^{9}+\frac{1203}{8}a^{8}-\frac{119}{4}a^{7}+\frac{217}{4}a^{6}+\frac{199}{2}a^{5}-\frac{357}{2}a^{4}+\frac{281}{4}a^{3}-117a^{2}+\frac{19}{2}a-\frac{41}{4}$, $\frac{11}{4}a^{15}-\frac{7}{4}a^{14}+\frac{161}{8}a^{13}-\frac{51}{4}a^{12}-\frac{21}{8}a^{11}+2a^{10}-\frac{1563}{8}a^{9}+\frac{989}{8}a^{8}-\frac{343}{4}a^{7}+\frac{101}{2}a^{6}+229a^{5}-\frac{567}{4}a^{4}+\frac{653}{4}a^{3}-97a^{2}+\frac{61}{4}a-\frac{33}{4}$
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| Regulator: | \( 94159.34556737287 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 94159.34556737287 \cdot 1}{2\cdot\sqrt{30257271966902092038144}}\cr\approx \mathstrut & 0.266451239004827 \end{aligned}\]
Galois group
$(C_2^3\times C_4):C_4$ (as 16T292):
| A solvable group of order 128 |
| The 26 conjugacy class representatives for $(C_2^3\times C_4):C_4$ |
| Character table for $(C_2^3\times C_4):C_4$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.18432.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.62h1.2064 | $x^{16} + 8 x^{15} + 20 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 30$ | $16$ | $1$ | $62$ | 16T292 | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |