Normalized defining polynomial
\( x^{16} - 8x^{12} - 10x^{10} + 5x^{8} + 26x^{6} + 32x^{4} + 14x^{2} + 1 \)
Invariants
| Degree: | $16$ |
| |
| Signature: | $[4, 6]$ |
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| Discriminant: |
\(27548985049130991616\)
\(\medspace = 2^{32}\cdot 283^{4}\)
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| Root discriminant: | \(16.41\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(283\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{19}{2}a^{14}-14a^{12}-55a^{10}-\frac{29}{2}a^{8}+67a^{6}+\frac{295}{2}a^{4}+\frac{177}{2}a^{2}+\frac{15}{2}$, $5a^{14}-\frac{15}{2}a^{12}-\frac{57}{2}a^{10}-\frac{15}{2}a^{8}+\frac{69}{2}a^{6}+\frac{155}{2}a^{4}+\frac{91}{2}a^{2}+5$, $a$, $a^{15}-2a^{13}-5a^{11}+\frac{3}{2}a^{9}+8a^{7}+\frac{23}{2}a^{5}-\frac{7}{2}a$, $\frac{7}{2}a^{15}-\frac{11}{2}a^{13}-\frac{39}{2}a^{11}-4a^{9}+\frac{49}{2}a^{7}+52a^{5}+28a^{3}+\frac{7}{2}a$, $10a^{14}-\frac{31}{2}a^{12}-\frac{113}{2}a^{10}-\frac{23}{2}a^{8}+\frac{141}{2}a^{6}+\frac{301}{2}a^{4}+\frac{167}{2}a^{2}-a+5$, $5a^{15}+\frac{1}{2}a^{14}-8a^{13}-a^{12}-28a^{11}-\frac{5}{2}a^{10}-4a^{9}+\frac{1}{2}a^{8}+36a^{7}+\frac{9}{2}a^{6}+73a^{5}+\frac{13}{2}a^{4}+38a^{3}-\frac{3}{2}$, $10a^{15}+\frac{15}{2}a^{14}-15a^{13}-\frac{23}{2}a^{12}-57a^{11}-\frac{85}{2}a^{10}-15a^{9}-\frac{19}{2}a^{8}+69a^{7}+\frac{105}{2}a^{6}+155a^{5}+\frac{229}{2}a^{4}+92a^{3}+64a^{2}+10a+4$, $6a^{15}+4a^{14}-\frac{19}{2}a^{13}-6a^{12}-\frac{67}{2}a^{11}-23a^{10}-6a^{9}-\frac{11}{2}a^{8}+\frac{85}{2}a^{7}+28a^{6}+89a^{5}+\frac{123}{2}a^{4}+\frac{93}{2}a^{3}+37a^{2}+\frac{3}{2}a+\frac{7}{2}$
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| Regulator: | \( 1940.67008057 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 1940.67008057 \cdot 1}{2\cdot\sqrt{27548985049130991616}}\cr\approx \mathstrut & 0.181998624553 \end{aligned}\]
Galois group
$C_4^3:(C_2\times S_4)$ (as 16T1535):
| A solvable group of order 3072 |
| The 45 conjugacy class representatives for $C_4^3:(C_2\times S_4)$ |
| Character table for $C_4^3:(C_2\times S_4)$ |
Intermediate fields
| 4.2.283.1, 8.4.20502784.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.0.27548985049130991616.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.32b54.3 | $x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 18 x^{12} + 20 x^{11} + 12 x^{10} + 32 x^{9} + 48 x^{8} + 36 x^{7} + 32 x^{6} + 52 x^{5} + 41 x^{4} + 24 x^{3} + 32 x^{2} + 20 x + 7$ | $4$ | $4$ | $32$ | 16T518 | $$[2, 2, 2, 3, 3, 3]^{4}$$ |
|
\(283\)
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |