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Properties

Label 16.0.76383686320...8576.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{10}\cdot 7^{6}$
Root discriminant $23.32$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 52, 212, 568, 980, 1028, 620, 144, -111, -68, 96, 56, 16, -16, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 16*x^13 + 16*x^12 + 56*x^11 + 96*x^10 - 68*x^9 - 111*x^8 + 144*x^7 + 620*x^6 + 1028*x^5 + 980*x^4 + 568*x^3 + 212*x^2 + 52*x + 7)
 
gp: K = bnfinit(x^16 - 8*x^14 - 16*x^13 + 16*x^12 + 56*x^11 + 96*x^10 - 68*x^9 - 111*x^8 + 144*x^7 + 620*x^6 + 1028*x^5 + 980*x^4 + 568*x^3 + 212*x^2 + 52*x + 7, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 16 x^{13} + 16 x^{12} + 56 x^{11} + 96 x^{10} - 68 x^{9} - 111 x^{8} + 144 x^{7} + 620 x^{6} + 1028 x^{5} + 980 x^{4} + 568 x^{3} + 212 x^{2} + 52 x + 7 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7638368632008213528576=2^{40}\cdot 3^{10}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.32$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{133} a^{14} - \frac{5}{19} a^{13} + \frac{1}{7} a^{11} + \frac{16}{133} a^{10} + \frac{47}{133} a^{9} - \frac{1}{19} a^{8} + \frac{5}{19} a^{7} + \frac{1}{133} a^{6} - \frac{59}{133} a^{5} + \frac{5}{133} a^{4} + \frac{2}{7} a^{3} - \frac{51}{133} a^{2} - \frac{3}{133} a + \frac{1}{19}$, $\frac{1}{103306836138779} a^{15} + \frac{189344810558}{103306836138779} a^{14} - \frac{186913505033}{776743128863} a^{13} + \frac{2649580135245}{5437201902041} a^{12} + \frac{4687256419876}{14758119448397} a^{11} + \frac{2144023521507}{14758119448397} a^{10} + \frac{39070578454220}{103306836138779} a^{9} - \frac{397258329408}{14758119448397} a^{8} + \frac{35514060372162}{103306836138779} a^{7} - \frac{41774200325944}{103306836138779} a^{6} + \frac{44027072664255}{103306836138779} a^{5} - \frac{590538109844}{5437201902041} a^{4} - \frac{40987257553206}{103306836138779} a^{3} - \frac{6153465797673}{103306836138779} a^{2} - \frac{7530388992661}{103306836138779} a + \frac{361540863116}{776743128863}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{940314496}{1509914441} a^{15} + \frac{65166208}{215702063} a^{14} + \frac{1051824008}{215702063} a^{13} + \frac{11420718600}{1509914441} a^{12} - \frac{21036763080}{1509914441} a^{11} - \frac{43046717080}{1509914441} a^{10} - \frac{9694675468}{215702063} a^{9} + \frac{14151004859}{215702063} a^{8} + \frac{60238860400}{1509914441} a^{7} - \frac{172415482660}{1509914441} a^{6} - \frac{500041782908}{1509914441} a^{5} - \frac{714174603932}{1509914441} a^{4} - \frac{545195424280}{1509914441} a^{3} - \frac{231975385804}{1509914441} a^{2} - \frac{8879455212}{215702063} a - \frac{1328450395}{215702063} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 119972.966475 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), 4.0.4032.1, 4.0.1008.2, \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.260112384.3

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$