Normalized defining polynomial
\( x^{16} - 8x^{14} + 52x^{12} - 408x^{10} + 1822x^{8} - 4024x^{6} + 4500x^{4} - 2472x^{2} + 289 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[8, 4]$ |
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| Discriminant: |
\(484116351470433472610304\)
\(\medspace = 2^{66}\cdot 3^{8}\)
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| Root discriminant: | \(30.22\) |
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| Galois root discriminant: | $2^{67/16}3^{1/2}\approx 31.559036391689393$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2\times C_4$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{34}a^{9}+\frac{4}{17}a^{7}+\frac{5}{17}a^{5}+\frac{2}{17}a^{3}+\frac{9}{34}a$, $\frac{1}{34}a^{10}+\frac{4}{17}a^{8}-\frac{7}{34}a^{6}-\frac{13}{34}a^{4}-\frac{4}{17}a^{2}-\frac{1}{2}$, $\frac{1}{68}a^{11}-\frac{1}{68}a^{10}-\frac{1}{68}a^{9}+\frac{9}{68}a^{8}+\frac{3}{34}a^{7}-\frac{5}{34}a^{6}-\frac{9}{34}a^{5}+\frac{15}{34}a^{4}-\frac{27}{68}a^{3}-\frac{9}{68}a^{2}-\frac{13}{68}a+\frac{1}{4}$, $\frac{1}{68}a^{12}-\frac{3}{68}a^{8}-\frac{2}{17}a^{6}+\frac{11}{68}a^{4}-\frac{1}{17}a^{2}-\frac{1}{4}$, $\frac{1}{68}a^{13}-\frac{1}{68}a^{9}+\frac{2}{17}a^{7}+\frac{31}{68}a^{5}+\frac{1}{17}a^{3}+\frac{1}{68}a$, $\frac{1}{519044}a^{14}+\frac{1561}{519044}a^{12}+\frac{6701}{519044}a^{10}+\frac{71517}{519044}a^{8}-\frac{69435}{519044}a^{6}-\frac{223087}{519044}a^{4}+\frac{213569}{519044}a^{2}+\frac{629}{1796}$, $\frac{1}{519044}a^{15}+\frac{1561}{519044}a^{13}-\frac{233}{129761}a^{11}-\frac{1}{68}a^{10}+\frac{705}{129761}a^{9}+\frac{9}{68}a^{8}+\frac{52693}{519044}a^{7}-\frac{5}{34}a^{6}-\frac{70427}{519044}a^{5}+\frac{15}{34}a^{4}-\frac{72591}{259522}a^{3}-\frac{9}{68}a^{2}-\frac{4307}{15266}a+\frac{1}{4}$
| 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: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{105}{129761}a^{14}+\frac{409}{259522}a^{12}-\frac{1369}{129761}a^{10}+\frac{16853}{129761}a^{8}+\frac{24059}{129761}a^{6}-\frac{758751}{259522}a^{4}+\frac{626915}{129761}a^{2}-\frac{491}{449}$, $\frac{5781}{519044}a^{14}+\frac{5241}{129761}a^{12}-\frac{191831}{519044}a^{10}+\frac{725969}{259522}a^{8}-\frac{3587979}{519044}a^{6}+\frac{771408}{129761}a^{4}-\frac{264161}{519044}a^{2}-\frac{797}{898}$, $\frac{5017}{519044}a^{15}+\frac{30453}{519044}a^{13}-\frac{216909}{519044}a^{11}+\frac{1690535}{519044}a^{9}-\frac{6395913}{519044}a^{7}+\frac{11847105}{519044}a^{5}-\frac{10977185}{519044}a^{3}+\frac{236811}{30532}a-1$, $\frac{756}{129761}a^{15}+\frac{2999}{129761}a^{13}-\frac{48719}{259522}a^{11}+\frac{196145}{129761}a^{9}-\frac{967953}{259522}a^{7}+\frac{516685}{259522}a^{5}+\frac{162978}{129761}a^{3}-\frac{18633}{15266}a+1$, $\frac{2073}{519044}a^{14}+\frac{2018}{129761}a^{12}-\frac{75443}{519044}a^{10}+\frac{275297}{259522}a^{8}-\frac{1561491}{519044}a^{6}+\frac{610335}{129761}a^{4}-\frac{2029649}{519044}a^{2}-\frac{1133}{898}$, $\frac{3715}{519044}a^{15}-\frac{2073}{519044}a^{14}+\frac{17231}{519044}a^{13}+\frac{2018}{129761}a^{12}-\frac{132763}{519044}a^{11}-\frac{75443}{519044}a^{10}+\frac{1057163}{519044}a^{9}+\frac{275297}{259522}a^{8}-\frac{3143373}{519044}a^{7}-\frac{1561491}{519044}a^{6}+\frac{3701969}{519044}a^{5}+\frac{610335}{129761}a^{4}-\frac{2614769}{519044}a^{3}-\frac{2029649}{519044}a^{2}+\frac{117969}{30532}a-\frac{1133}{898}$, $\frac{811}{519044}a^{15}+\frac{2073}{519044}a^{14}+\frac{1107}{519044}a^{13}-\frac{2018}{129761}a^{12}-\frac{15081}{519044}a^{11}+\frac{75443}{519044}a^{10}+\frac{132641}{519044}a^{9}-\frac{275297}{259522}a^{8}+\frac{117639}{519044}a^{7}+\frac{1561491}{519044}a^{6}-\frac{1749399}{519044}a^{5}-\frac{610335}{129761}a^{4}+\frac{2690393}{519044}a^{3}+\frac{2029649}{519044}a^{2}-\frac{88041}{30532}a-\frac{663}{898}$, $\frac{1457}{129761}a^{15}-\frac{7890}{129761}a^{13}+\frac{115995}{259522}a^{11}-\frac{455794}{129761}a^{9}+\frac{3139137}{259522}a^{7}-\frac{5048853}{259522}a^{5}+\frac{2071698}{129761}a^{3}-\frac{89651}{15266}a$, $\frac{2715}{259522}a^{15}-\frac{6733}{129761}a^{13}+\frac{103005}{259522}a^{11}-\frac{400532}{129761}a^{9}+\frac{1295698}{129761}a^{7}-\frac{3635963}{259522}a^{5}+\frac{1022317}{129761}a^{3}-\frac{17907}{15266}a$, $\frac{1771}{259522}a^{15}-\frac{166}{129761}a^{14}+\frac{35395}{519044}a^{13}-\frac{6049}{519044}a^{12}-\frac{52509}{129761}a^{11}+\frac{2052}{129761}a^{10}+\frac{1682417}{519044}a^{9}-\frac{124523}{519044}a^{8}-\frac{4120065}{259522}a^{7}+\frac{863289}{259522}a^{6}+\frac{18745009}{519044}a^{5}-\frac{5148505}{519044}a^{4}-\frac{5057004}{129761}a^{3}+\frac{3280219}{259522}a^{2}+\frac{578203}{30532}a-\frac{13107}{1796}$, $\frac{2253}{519044}a^{15}+\frac{7071}{259522}a^{14}-\frac{8573}{259522}a^{13}-\frac{67671}{519044}a^{12}+\frac{106141}{519044}a^{11}+\frac{132131}{129761}a^{10}-\frac{433679}{259522}a^{9}-\frac{4108539}{519044}a^{8}+\frac{3626955}{519044}a^{7}+\frac{6452459}{259522}a^{6}-\frac{3368583}{259522}a^{5}-\frac{18061573}{519044}a^{4}+\frac{5680855}{519044}a^{3}+\frac{2928829}{129761}a^{2}-\frac{60049}{15266}a-\frac{3413}{1796}$
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| Regulator: | \( 1063863.751199081 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1063863.751199081 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.305028612196671 \end{aligned}\]
Galois group
$C_2^2:C_8$ (as 16T24):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.9216.1, \(\Q(\zeta_{16})^+\), 4.2.18432.3, 8.4.173946175488.1, \(\Q(\zeta_{32})^+\), 8.4.5435817984.1 |
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: | 16.0.484116351470433472610304.196 |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66h1.519 | $x^{16} + 16 x^{13} + 4 x^{12} + 8 x^{10} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | $C_2^2 : C_8$ | $$[2, 3, \frac{7}{2}, 4, 5]$$ |
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\(3\)
| 3.8.2.8a1.2 | $x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $$[\ ]_{2}^{8}$$ |