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Properties

Label 16.4.64964301804...6416.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{25}\cdot 13^{7}\cdot 17^{8}\cdot 89^{7}$
Root discriminant $266.56$
Ramified primes $2, 13, 17, 89$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2369536, 0, -13328640, 0, 11264552, 0, 7832890, 0, 1620865, 0, 151178, 0, 6730, 0, 140, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 140*x^14 + 6730*x^12 + 151178*x^10 + 1620865*x^8 + 7832890*x^6 + 11264552*x^4 - 13328640*x^2 + 2369536)
 
gp: K = bnfinit(x^16 + 140*x^14 + 6730*x^12 + 151178*x^10 + 1620865*x^8 + 7832890*x^6 + 11264552*x^4 - 13328640*x^2 + 2369536, 1)
 

Normalized defining polynomial

\( x^{16} + 140 x^{14} + 6730 x^{12} + 151178 x^{10} + 1620865 x^{8} + 7832890 x^{6} + 11264552 x^{4} - 13328640 x^{2} + 2369536 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(649643018047387088671190961146262716416=2^{25}\cdot 13^{7}\cdot 17^{8}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $266.56$
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:  $2, 13, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{8} a^{11} - \frac{3}{16} a^{9} - \frac{3}{16} a^{7} - \frac{15}{32} a^{5} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{617705061844613210633728} a^{14} - \frac{1}{64} a^{13} - \frac{2953297800016563569881}{154426265461153302658432} a^{12} + \frac{1}{16} a^{11} - \frac{61772634842111574740731}{308852530922306605316864} a^{10} - \frac{5}{32} a^{9} + \frac{50853744566413854729621}{308852530922306605316864} a^{8} - \frac{5}{32} a^{7} + \frac{69305566680512408333345}{617705061844613210633728} a^{6} + \frac{31}{64} a^{5} - \frac{63332231469439334082651}{308852530922306605316864} a^{4} - \frac{5}{32} a^{3} - \frac{19167651582241120670759}{77213132730576651329216} a^{2} - \frac{1}{8} a - \frac{1357132437314242714557}{4825820795661040708076}$, $\frac{1}{2470820247378452842534912} a^{15} - \frac{2953297800016563569881}{617705061844613210633728} a^{13} + \frac{92653630619041727917701}{1235410123689226421267456} a^{11} - \frac{1}{4} a^{10} + \frac{50853744566413854729621}{1235410123689226421267456} a^{9} + \frac{69305566680512408333345}{2470820247378452842534912} a^{7} - \frac{372184762391745939399515}{1235410123689226421267456} a^{5} - \frac{1}{2} a^{4} - \frac{57774217947529446335367}{308852530922306605316864} a^{3} + \frac{1}{4} a^{2} - \frac{1357132437314242714557}{19303283182644162832304} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  $9$
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:  \( 2898621472490 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.334373.2, 8.4.16557918172192384.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.37$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$