Normalized defining polynomial
\( x^{16} - 4 x^{15} + 6 x^{14} - 4 x^{13} - 5 x^{12} + 12 x^{11} + 12 x^{10} - 68 x^{9} + 126 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2069703095939497984\) \(\medspace = 2^{44}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(13.96\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2^{11/4}7^{1/2}\approx 17.798422345016238$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{162122591}a^{15}+\frac{45424346}{162122591}a^{14}+\frac{49494767}{162122591}a^{13}+\frac{2043155}{162122591}a^{12}-\frac{34864797}{162122591}a^{11}+\frac{835662}{162122591}a^{10}+\frac{19712972}{162122591}a^{9}-\frac{11118121}{162122591}a^{8}+\frac{13623379}{162122591}a^{7}+\frac{19572739}{162122591}a^{6}-\frac{2662856}{9536623}a^{5}-\frac{21530626}{162122591}a^{4}+\frac{52787661}{162122591}a^{3}+\frac{26703603}{162122591}a^{2}+\frac{40710260}{162122591}a+\frac{31377500}{162122591}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{311497354}{162122591}a^{15}-\frac{1033906602}{162122591}a^{14}+\frac{1165489544}{162122591}a^{13}-\frac{470028008}{162122591}a^{12}-\frac{1838666762}{162122591}a^{11}+\frac{2469728748}{162122591}a^{10}+\frac{5409432464}{162122591}a^{9}-\frac{17399288206}{162122591}a^{8}+\frac{27382010314}{162122591}a^{7}-\frac{22859545403}{162122591}a^{6}+\frac{402173188}{9536623}a^{5}-\frac{3532601944}{162122591}a^{4}+\frac{6536830308}{162122591}a^{3}-\frac{6565540449}{162122591}a^{2}+\frac{3233734820}{162122591}a-\frac{604386706}{162122591}$, $\frac{177713584}{162122591}a^{15}-\frac{802331092}{162122591}a^{14}+\frac{1295069240}{162122591}a^{13}-\frac{814725266}{162122591}a^{12}-\frac{996409842}{162122591}a^{11}+\frac{2736573289}{162122591}a^{10}+\frac{1879308492}{162122591}a^{9}-\frac{14204316824}{162122591}a^{8}+\frac{26113970486}{162122591}a^{7}-\frac{27451339622}{162122591}a^{6}+\frac{775656064}{9536623}a^{5}-\frac{1881192114}{162122591}a^{4}+\frac{5312423196}{162122591}a^{3}-\frac{7933924910}{162122591}a^{2}+\frac{4731903908}{162122591}a-\frac{967323865}{162122591}$, $\frac{309954798}{162122591}a^{15}-\frac{1085422187}{162122591}a^{14}+\frac{1269587313}{162122591}a^{13}-\frac{487863148}{162122591}a^{12}-\frac{1863978060}{162122591}a^{11}+\frac{2775262899}{162122591}a^{10}+\frac{5407854356}{162122591}a^{9}-\frac{18510300593}{162122591}a^{8}+\frac{28832710102}{162122591}a^{7}-\frac{24598752858}{162122591}a^{6}+\frac{411487810}{9536623}a^{5}-\frac{2959661193}{162122591}a^{4}+\frac{7603165180}{162122591}a^{3}-\frac{7071261392}{162122591}a^{2}+\frac{3239288928}{162122591}a-\frac{617856247}{162122591}$, $\frac{91476756}{162122591}a^{15}-\frac{228787736}{162122591}a^{14}+\frac{103870930}{162122591}a^{13}+\frac{107841922}{162122591}a^{12}-\frac{604010281}{162122591}a^{11}+\frac{253254516}{162122591}a^{10}+\frac{2119793112}{162122591}a^{9}-\frac{3722058902}{162122591}a^{8}+\frac{3993146534}{162122591}a^{7}-\frac{746202208}{162122591}a^{6}-\frac{140231406}{9536623}a^{5}-\frac{336442844}{162122591}a^{4}+\frac{1536235886}{162122591}a^{3}-\frac{466777892}{162122591}a^{2}-\frac{292262216}{162122591}a+\frac{177713584}{162122591}$, $\frac{272000763}{162122591}a^{15}-\frac{893172328}{162122591}a^{14}+\frac{961050838}{162122591}a^{13}-\frac{310761635}{162122591}a^{12}-\frac{1641392537}{162122591}a^{11}+\frac{2072944059}{162122591}a^{10}+\frac{4932967058}{162122591}a^{9}-\frac{15095690477}{162122591}a^{8}+\frac{22749698818}{162122591}a^{7}-\frac{18178841118}{162122591}a^{6}+\frac{252671424}{9536623}a^{5}-\frac{2558897871}{162122591}a^{4}+\frac{6067696677}{162122591}a^{3}-\frac{4906796723}{162122591}a^{2}+\frac{2066044878}{162122591}a-\frac{244973002}{162122591}$, $\frac{311497354}{162122591}a^{15}-\frac{1033906602}{162122591}a^{14}+\frac{1165489544}{162122591}a^{13}-\frac{470028008}{162122591}a^{12}-\frac{1838666762}{162122591}a^{11}+\frac{2469728748}{162122591}a^{10}+\frac{5409432464}{162122591}a^{9}-\frac{17399288206}{162122591}a^{8}+\frac{27382010314}{162122591}a^{7}-\frac{22859545403}{162122591}a^{6}+\frac{402173188}{9536623}a^{5}-\frac{3532601944}{162122591}a^{4}+\frac{6536830308}{162122591}a^{3}-\frac{6565540449}{162122591}a^{2}+\frac{3395857411}{162122591}a-\frac{604386706}{162122591}$, $\frac{387525287}{162122591}a^{15}-\frac{1445117866}{162122591}a^{14}+\frac{1821846952}{162122591}a^{13}-\frac{730655679}{162122591}a^{12}-\frac{2411095003}{162122591}a^{11}+\frac{4009682087}{162122591}a^{10}+\frac{6440528662}{162122591}a^{9}-\frac{25204883387}{162122591}a^{8}+\frac{39750495998}{162122591}a^{7}-\frac{34917555532}{162122591}a^{6}+\frac{608692943}{9536623}a^{5}-\frac{1960310423}{162122591}a^{4}+\frac{10547431755}{162122591}a^{3}-\frac{10420449738}{162122591}a^{2}+\frac{4209270063}{162122591}a-\frac{782368941}{162122591}$, $\frac{182242031}{162122591}a^{15}-\frac{526666412}{162122591}a^{14}+\frac{431535634}{162122591}a^{13}-\frac{21250396}{162122591}a^{12}-\frac{1134422160}{162122591}a^{11}+\frac{938782580}{162122591}a^{10}+\frac{3735656920}{162122591}a^{9}-\frac{8694711403}{162122591}a^{8}+\frac{11763146479}{162122591}a^{7}-\frac{7116535024}{162122591}a^{6}-\frac{25659304}{9536623}a^{5}-\frac{1653682536}{162122591}a^{4}+\frac{3622557255}{162122591}a^{3}-\frac{2514508447}{162122591}a^{2}+\frac{309578932}{162122591}a+\frac{141130368}{162122591}$, $\frac{397142741}{162122591}a^{15}-\frac{1282904979}{162122591}a^{14}+\frac{1328739203}{162122591}a^{13}-\frac{363702691}{162122591}a^{12}-\frac{2457258318}{162122591}a^{11}+\frac{2912464855}{162122591}a^{10}+\frac{7424923231}{162122591}a^{9}-\frac{21700576269}{162122591}a^{8}+\frac{32061467999}{162122591}a^{7}-\frac{24354520347}{162122591}a^{6}+\frac{246569043}{9536623}a^{5}-\frac{2955818151}{162122591}a^{4}+\frac{8500621441}{162122591}a^{3}-\frac{7332753121}{162122591}a^{2}+\frac{2665450555}{162122591}a-\frac{219429157}{162122591}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 586.589293907 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 586.589293907 \cdot 1}{2\cdot\sqrt{2069703095939497984}}\cr\approx \mathstrut & 0.200700892939 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 16T28):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.2.448.1, 4.2.14336.1, 8.4.205520896.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.0.4917248.1, 8.0.314703872.3 |
Degree 16 sibling: | 16.0.99038527051792384.1 |
Minimal sibling: | 8.0.4917248.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $44$ | |||
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |