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Properties

Label 16.0.25316504136...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{10}\cdot 11^{6}\cdot 151^{4}$
Root discriminant $188.46$
Ramified primes $2, 5, 11, 151$
Class number $18230400$ (GRH)
Class group $[2, 2, 2, 2, 2, 569700]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25162463088400, 0, 6665553136000, 0, 653031584480, 0, 32138356800, 0, 881319704, 0, 13877280, 0, 122936, 0, 560, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 560*x^14 + 122936*x^12 + 13877280*x^10 + 881319704*x^8 + 32138356800*x^6 + 653031584480*x^4 + 6665553136000*x^2 + 25162463088400)
 
gp: K = bnfinit(x^16 + 560*x^14 + 122936*x^12 + 13877280*x^10 + 881319704*x^8 + 32138356800*x^6 + 653031584480*x^4 + 6665553136000*x^2 + 25162463088400, 1)
 

Normalized defining polynomial

\( x^{16} + 560 x^{14} + 122936 x^{12} + 13877280 x^{10} + 881319704 x^{8} + 32138356800 x^{6} + 653031584480 x^{4} + 6665553136000 x^{2} + 25162463088400 \)

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:  \(2531650413679670693032099840000000000=2^{48}\cdot 5^{10}\cdot 11^{6}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.46$
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:  $2, 5, 11, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{224} a^{8} - \frac{5}{56} a^{4} - \frac{3}{7} a^{2} + \frac{5}{56}$, $\frac{1}{224} a^{9} - \frac{5}{56} a^{5} - \frac{3}{7} a^{3} + \frac{5}{56} a$, $\frac{1}{372064} a^{10} - \frac{1}{8456} a^{8} - \frac{66}{1057} a^{6} - \frac{155}{6644} a^{4} - \frac{3163}{8456} a^{2} + \frac{2}{7}$, $\frac{1}{744128} a^{11} - \frac{1}{16912} a^{9} + \frac{529}{16912} a^{7} - \frac{155}{13288} a^{5} - \frac{1049}{16912} a^{3} - \frac{5}{14} a$, $\frac{1}{1123633280} a^{12} + \frac{1}{2006488} a^{10} - \frac{108417}{51074240} a^{8} - \frac{578967}{14045416} a^{6} - \frac{31735129}{280908320} a^{4} + \frac{583}{4228} a^{2} - \frac{43}{112}$, $\frac{1}{2247266560} a^{13} + \frac{1}{4012976} a^{11} + \frac{119593}{102148480} a^{9} + \frac{588355}{14045416} a^{7} + \frac{13410851}{561816640} a^{5} - \frac{43}{2114} a^{3} + \frac{23}{224} a$, $\frac{1}{24581283481359719699200} a^{14} - \frac{24236643943}{122906417406798598496} a^{12} - \frac{621216288442437}{12290641740679859849600} a^{10} - \frac{48348575245427157}{614532087033992992480} a^{8} + \frac{10785076027747753173}{877902981477132846400} a^{6} + \frac{26535832477045943}{508718615094365060} a^{4} + \frac{320605888780671}{2450179964331680} a^{2} - \frac{39075256021}{811317868984}$, $\frac{1}{24581283481359719699200} a^{15} - \frac{24236643943}{122906417406798598496} a^{13} - \frac{621216288442437}{12290641740679859849600} a^{11} - \frac{48348575245427157}{614532087033992992480} a^{9} + \frac{10785076027747753173}{877902981477132846400} a^{7} + \frac{26535832477045943}{508718615094365060} a^{5} + \frac{320605888780671}{2450179964331680} a^{3} - \frac{39075256021}{811317868984} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{569700}$, which has order $18230400$ (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:  \( -1 \) (order $2$)
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:  \( 17367.3059334 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.17600.1, 4.4.4400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.4956160000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
151Data not computed