Normalized defining polynomial
\( x^{16} + 2 x^{14} - 4 x^{12} - 12 x^{11} + 36 x^{10} - 52 x^{9} + 83 x^{8} - 88 x^{7} + 60 x^{6} - 56 x^{5} + 44 x^{4} - 16 x^{3} + 6 x^{2} - 4 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(281792804290560000\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 5^{4}\) | 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: | \(12.32\) | 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: | $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55}a^{14}-\frac{16}{55}a^{13}-\frac{9}{55}a^{12}+\frac{16}{55}a^{11}-\frac{2}{55}a^{10}-\frac{17}{55}a^{9}+\frac{17}{55}a^{8}-\frac{4}{11}a^{7}-\frac{1}{5}a^{6}-\frac{17}{55}a^{5}+\frac{24}{55}a^{4}-\frac{26}{55}a^{3}-\frac{8}{55}a^{2}+\frac{14}{55}a-\frac{7}{55}$, $\frac{1}{5383015}a^{15}+\frac{45429}{5383015}a^{14}+\frac{1218441}{5383015}a^{13}+\frac{2340626}{5383015}a^{12}-\frac{1235062}{5383015}a^{11}+\frac{1589068}{5383015}a^{10}-\frac{82243}{5383015}a^{9}-\frac{315873}{1076603}a^{8}+\frac{2408909}{5383015}a^{7}-\frac{899982}{5383015}a^{6}-\frac{214536}{489365}a^{5}-\frac{763176}{5383015}a^{4}+\frac{2068647}{5383015}a^{3}+\frac{1258899}{5383015}a^{2}-\frac{2347857}{5383015}a+\frac{416120}{1076603}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{7335514}{1076603} a^{15} + \frac{21771183}{5383015} a^{14} + \frac{88719952}{5383015} a^{13} + \frac{54921153}{5383015} a^{12} - \frac{107077247}{5383015} a^{11} - \frac{496850306}{5383015} a^{10} + \frac{1021962669}{5383015} a^{9} - \frac{1332800764}{5383015} a^{8} + \frac{462145676}{1076603} a^{7} - \frac{1932399318}{5383015} a^{6} + \frac{107675024}{489365} a^{5} - \frac{1432973573}{5383015} a^{4} + \frac{817209282}{5383015} a^{3} - \frac{168754859}{5383015} a^{2} + \frac{144507767}{5383015} a - \frac{68578216}{5383015} \) (order $24$) | 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: | $\frac{28548856}{5383015}a^{15}+\frac{3078458}{1076603}a^{14}+\frac{1204568}{97873}a^{13}+\frac{37307912}{5383015}a^{12}-\frac{8266331}{489365}a^{11}-\frac{387358399}{5383015}a^{10}+\frac{163829922}{1076603}a^{9}-\frac{96060953}{489365}a^{8}+\frac{1808031804}{5383015}a^{7}-\frac{1544640188}{5383015}a^{6}+\frac{905840102}{5383015}a^{5}-\frac{1100844652}{5383015}a^{4}+\frac{656515941}{5383015}a^{3}-\frac{116181614}{5383015}a^{2}+\frac{100583072}{5383015}a-\frac{53763817}{5383015}$, $\frac{21771183}{5383015}a^{15}+\frac{15364812}{5383015}a^{14}+\frac{54921153}{5383015}a^{13}+\frac{3603003}{489365}a^{12}-\frac{56719466}{5383015}a^{11}-\frac{298429851}{5383015}a^{10}+\frac{574432876}{5383015}a^{9}-\frac{146701986}{1076603}a^{8}+\frac{1295226842}{5383015}a^{7}-\frac{1016228936}{5383015}a^{6}+\frac{620970347}{5383015}a^{5}-\frac{796603798}{5383015}a^{4}+\frac{418086261}{5383015}a^{3}-\frac{75557653}{5383015}a^{2}+\frac{78132064}{5383015}a-\frac{7335514}{1076603}$, $\frac{10508290}{1076603}a^{15}+\frac{7408497}{1076603}a^{14}+\frac{26524677}{1076603}a^{13}+\frac{19065077}{1076603}a^{12}-\frac{27508077}{1076603}a^{11}-\frac{144248466}{1076603}a^{10}+\frac{276957070}{1076603}a^{9}-\frac{354610539}{1076603}a^{8}+\frac{627314593}{1076603}a^{7}-\frac{491222859}{1076603}a^{6}+\frac{300378937}{1076603}a^{5}-\frac{35088703}{97873}a^{4}+\frac{202376257}{1076603}a^{3}-\frac{36565289}{1076603}a^{2}+\frac{40461981}{1076603}a-\frac{17757919}{1076603}$, $\frac{14903225}{1076603}a^{15}+\frac{7611913}{1076603}a^{14}+\frac{33467510}{1076603}a^{13}+\frac{17048809}{1076603}a^{12}-\frac{51301040}{1076603}a^{11}-\frac{205104761}{1076603}a^{10}+\frac{432754775}{1076603}a^{9}-\frac{550959267}{1076603}a^{8}+\frac{947769389}{1076603}a^{7}-\frac{817918154}{1076603}a^{6}+\frac{462016375}{1076603}a^{5}-\frac{53099979}{97873}a^{4}+\frac{349781551}{1076603}a^{3}-\frac{52038528}{1076603}a^{2}+\frac{57040933}{1076603}a-\frac{29287961}{1076603}$, $\frac{575386}{5383015}a^{15}+\frac{5945118}{5383015}a^{14}+\frac{7343147}{5383015}a^{13}+\frac{3704353}{1076603}a^{12}+\frac{16315862}{5383015}a^{11}-\frac{2492058}{1076603}a^{10}-\frac{58696971}{5383015}a^{9}+\frac{96925508}{5383015}a^{8}-\frac{131593696}{5383015}a^{7}+\frac{274920889}{5383015}a^{6}-\frac{163323474}{5383015}a^{5}+\frac{25218667}{1076603}a^{4}-\frac{14282947}{489365}a^{3}+\frac{41863927}{5383015}a^{2}-\frac{22471621}{5383015}a+\frac{15882322}{5383015}$, $\frac{44958898}{5383015}a^{15}+\frac{4626477}{1076603}a^{14}+\frac{1855954}{97873}a^{13}+\frac{51904186}{5383015}a^{12}-\frac{13908363}{489365}a^{11}-\frac{619850747}{5383015}a^{10}+\frac{259459137}{1076603}a^{9}-\frac{152023424}{489365}a^{8}+\frac{2888037137}{5383015}a^{7}-\frac{2501099769}{5383015}a^{6}+\frac{1455860826}{5383015}a^{5}-\frac{1830800516}{5383015}a^{4}+\frac{1082490303}{5383015}a^{3}-\frac{198687312}{5383015}a^{2}+\frac{201199071}{5383015}a-\frac{92103701}{5383015}$, $\frac{68210074}{5383015}a^{15}+\frac{41492923}{5383015}a^{14}+\frac{165086467}{5383015}a^{13}+\frac{103560326}{5383015}a^{12}-\frac{200235361}{5383015}a^{11}-\frac{930987672}{5383015}a^{10}+\frac{1883578234}{5383015}a^{9}-\frac{2446934516}{5383015}a^{8}+\frac{386753806}{489365}a^{7}-\frac{703954011}{1076603}a^{6}+\frac{2129837717}{5383015}a^{5}-\frac{2631223961}{5383015}a^{4}+\frac{1467644821}{5383015}a^{3}-\frac{56709729}{1076603}a^{2}+\frac{52403763}{1076603}a-\frac{10652134}{489365}$ | 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: | \( 1291.96804794 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1291.96804794 \cdot 1}{24\cdot\sqrt{281792804290560000}}\cr\approx \mathstrut & 0.246328424291 \end{aligned}\]
Galois group
$C_2^2\times D_4$ (as 16T25):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
Character table for $C_2^2 \times D_4$ |
Intermediate fields
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 siblings: | 16.0.11007531417600000000.6, 16.0.11007531417600000000.9, 16.0.176120502681600000000.4, 16.8.176120502681600000000.2, 16.0.176120502681600000000.5, 16.0.176120502681600000000.7, 16.0.176120502681600000000.10 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 $16$ | $8$ | $2$ | $36$ | |||
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |