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Properties

Label 16.0.17631537627...1024.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 37^{6}$
Root discriminant $18.42$
Ramified primes $2, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 212, 594, 1056, 1180, 812, 320, 100, 170, 24, 66, -24, 54, -32, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 32*x^13 + 54*x^12 - 24*x^11 + 66*x^10 + 24*x^9 + 170*x^8 + 100*x^7 + 320*x^6 + 812*x^5 + 1180*x^4 + 1056*x^3 + 594*x^2 + 212*x + 41)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 32*x^13 + 54*x^12 - 24*x^11 + 66*x^10 + 24*x^9 + 170*x^8 + 100*x^7 + 320*x^6 + 812*x^5 + 1180*x^4 + 1056*x^3 + 594*x^2 + 212*x + 41, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 32 x^{13} + 54 x^{12} - 24 x^{11} + 66 x^{10} + 24 x^{9} + 170 x^{8} + 100 x^{7} + 320 x^{6} + 812 x^{5} + 1180 x^{4} + 1056 x^{3} + 594 x^{2} + 212 x + 41 \)

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:  \(176315376274216321024=2^{36}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.42$
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:  $2, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{99423960989000442583} a^{15} - \frac{28196119997392583202}{99423960989000442583} a^{14} - \frac{6907154079231693100}{99423960989000442583} a^{13} + \frac{47059612001237201641}{99423960989000442583} a^{12} - \frac{35162031974028465966}{99423960989000442583} a^{11} - \frac{12672746741825870909}{99423960989000442583} a^{10} + \frac{3716405784558271381}{99423960989000442583} a^{9} - \frac{20380802161089248975}{99423960989000442583} a^{8} - \frac{42501615183324095585}{99423960989000442583} a^{7} + \frac{41753954068161157668}{99423960989000442583} a^{6} - \frac{327343005295522021}{1483939716253737949} a^{5} + \frac{29083924244012652759}{99423960989000442583} a^{4} - \frac{18389989596279505152}{99423960989000442583} a^{3} + \frac{49483388735991993653}{99423960989000442583} a^{2} + \frac{18171390188513121351}{99423960989000442583} a + \frac{32621555456942289532}{99423960989000442583}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( \frac{55863394562}{855414899051} a^{15} - \frac{259899409805}{855414899051} a^{14} + \frac{1047330607788}{855414899051} a^{13} - \frac{2398426208153}{855414899051} a^{12} + \frac{4292830015556}{855414899051} a^{11} - \frac{3503220451138}{855414899051} a^{10} + \frac{4876300238990}{855414899051} a^{9} - \frac{1100119197745}{855414899051} a^{8} + \frac{9153189861366}{855414899051} a^{7} - \frac{435616414685}{855414899051} a^{6} + \frac{236875316226}{12767386553} a^{5} + \frac{34707068768061}{855414899051} a^{4} + \frac{39336082994276}{855414899051} a^{3} + \frac{22841051088690}{855414899051} a^{2} + \frac{7148955664728}{855414899051} a + \frac{965800620934}{855414899051} \) (order $4$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8796.0796071 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.592.1, 8.0.5607424.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$