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Properties

Label 16.8.31087867512...0961.3
Degree $16$
Signature $[8, 4]$
Discriminant $31^{10}\cdot 41^{14}$
Root discriminant $220.44$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15499969, 0, 43746642, 0, -80735935, 0, -8974035, 0, 1213530, 0, 20587, 0, -3233, 0, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 17*x^14 - 3233*x^12 + 20587*x^10 + 1213530*x^8 - 8974035*x^6 - 80735935*x^4 + 43746642*x^2 + 15499969)
 
gp: K = bnfinit(x^16 + 17*x^14 - 3233*x^12 + 20587*x^10 + 1213530*x^8 - 8974035*x^6 - 80735935*x^4 + 43746642*x^2 + 15499969, 1)
 

Normalized defining polynomial

\( x^{16} + 17 x^{14} - 3233 x^{12} + 20587 x^{10} + 1213530 x^{8} - 8974035 x^{6} - 80735935 x^{4} + 43746642 x^{2} + 15499969 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31087867512274251337747808514764310961=31^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.44$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{248} a^{12} + \frac{6}{31} a^{10} + \frac{11}{124} a^{8} - \frac{59}{248} a^{6} - \frac{1}{2} a^{5} + \frac{35}{248} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{248} a^{13} + \frac{6}{31} a^{11} + \frac{11}{124} a^{9} - \frac{59}{248} a^{7} + \frac{35}{248} a^{5} + \frac{3}{8} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{2130384164130577139356099888} a^{14} - \frac{1}{496} a^{13} + \frac{1985165673262120379522487}{2130384164130577139356099888} a^{12} - \frac{3}{31} a^{11} + \frac{49866448165913844655938669}{1065192082065288569678049944} a^{10} + \frac{51}{248} a^{9} + \frac{449917435129444697337034479}{2130384164130577139356099888} a^{8} - \frac{65}{496} a^{7} - \frac{14780358289409704223631285}{1065192082065288569678049944} a^{6} - \frac{159}{496} a^{5} + \frac{124969490878264484491329685}{1065192082065288569678049944} a^{4} + \frac{1}{16} a^{3} - \frac{90366368939190372105715}{291195211062134655461468} a^{2} + \frac{5}{16} a + \frac{10259257537838120616631629}{68722069810663778688906448}$, $\frac{1}{270558788844583296698224685776} a^{15} + \frac{3613865462151652535266186}{16909924302786456043639042861} a^{13} - \frac{1}{496} a^{12} + \frac{18965616163551118928777438481}{135279394422291648349112342888} a^{11} - \frac{3}{31} a^{10} + \frac{18188801705007032571411011369}{270558788844583296698224685776} a^{9} + \frac{51}{248} a^{8} - \frac{28355938866661795686780889105}{270558788844583296698224685776} a^{7} - \frac{65}{496} a^{6} + \frac{65583151724881950071292408153}{270558788844583296698224685776} a^{5} + \frac{89}{496} a^{4} - \frac{64206015501129784698349719}{147927167219564404974425744} a^{3} - \frac{7}{16} a^{2} + \frac{17553275463582921825262548}{545481429122143743343194931} a - \frac{3}{16}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 4127183437040 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
31Data not computed
41Data not computed