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Properties

Label 16.8.12136551081...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $56.92$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-229679, 1837308, -4781532, 4629640, -1910700, 4492, 480604, -307740, 102439, -15060, -3324, 1844, -440, 80, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 80*x^13 - 440*x^12 + 1844*x^11 - 3324*x^10 - 15060*x^9 + 102439*x^8 - 307740*x^7 + 480604*x^6 + 4492*x^5 - 1910700*x^4 + 4629640*x^3 - 4781532*x^2 + 1837308*x - 229679)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 80*x^13 - 440*x^12 + 1844*x^11 - 3324*x^10 - 15060*x^9 + 102439*x^8 - 307740*x^7 + 480604*x^6 + 4492*x^5 - 1910700*x^4 + 4629640*x^3 - 4781532*x^2 + 1837308*x - 229679, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 80 x^{13} - 440 x^{12} + 1844 x^{11} - 3324 x^{10} - 15060 x^{9} + 102439 x^{8} - 307740 x^{7} + 480604 x^{6} + 4492 x^{5} - 1910700 x^{4} + 4629640 x^{3} - 4781532 x^{2} + 1837308 x - 229679 \)

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:  \(12136551081312256000000000000=2^{44}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.92$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{6}{31} a^{13} + \frac{4}{31} a^{12} + \frac{10}{31} a^{11} + \frac{5}{31} a^{10} + \frac{5}{31} a^{9} + \frac{14}{31} a^{8} + \frac{9}{31} a^{7} - \frac{3}{31} a^{6} + \frac{3}{31} a^{5} + \frac{5}{31} a^{4} - \frac{13}{31} a^{3} + \frac{11}{31} a^{2} - \frac{15}{31} a$, $\frac{1}{673313264989433477314300667846413446855633293849} a^{15} - \frac{463406255102891535548608599037493608352252607}{673313264989433477314300667846413446855633293849} a^{14} - \frac{34641086571288024792994636188322301862633032539}{673313264989433477314300667846413446855633293849} a^{13} - \frac{3030897200152311562818583623818783878950723552}{35437540262601761963910561465600707729243857571} a^{12} + \frac{212660733378567300926721382533527381599593918493}{673313264989433477314300667846413446855633293849} a^{11} + \frac{313302616997179658937293735070751225686736583261}{673313264989433477314300667846413446855633293849} a^{10} - \frac{87489230960172242142917828172277543974359838446}{673313264989433477314300667846413446855633293849} a^{9} - \frac{112305350530928302862380608015592250708866988861}{673313264989433477314300667846413446855633293849} a^{8} + \frac{62701650088146689079918569607898484518525134365}{673313264989433477314300667846413446855633293849} a^{7} - \frac{112754782842425982792312271106464176260276997155}{673313264989433477314300667846413446855633293849} a^{6} - \frac{16634841864449419013428227648649196068013890914}{35437540262601761963910561465600707729243857571} a^{5} + \frac{41988812161518970723126602861617838024822248942}{673313264989433477314300667846413446855633293849} a^{4} + \frac{117522865921922152907284765946025365733524640439}{673313264989433477314300667846413446855633293849} a^{3} - \frac{195833785289431376246642555724635056321934612921}{673313264989433477314300667846413446855633293849} a^{2} - \frac{299496118880514608318617055580622829362732931030}{673313264989433477314300667846413446855633293849} a - \frac{5093479645117502376890428561735858841944665516}{21719782741594628300461311866013336995343009479}$

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:  \( 66685480.5243 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed