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Properties

Label 16.0.42299090063...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 101^{8}$
Root discriminant $22.47$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 125, -275, 1275, 365, -100, 199, -735, 449, -436, 282, -129, 89, -25, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 25*x^13 + 89*x^12 - 129*x^11 + 282*x^10 - 436*x^9 + 449*x^8 - 735*x^7 + 199*x^6 - 100*x^5 + 365*x^4 + 1275*x^3 - 275*x^2 + 125*x + 625)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 25*x^13 + 89*x^12 - 129*x^11 + 282*x^10 - 436*x^9 + 449*x^8 - 735*x^7 + 199*x^6 - 100*x^5 + 365*x^4 + 1275*x^3 - 275*x^2 + 125*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 25 x^{13} + 89 x^{12} - 129 x^{11} + 282 x^{10} - 436 x^{9} + 449 x^{8} - 735 x^{7} + 199 x^{6} - 100 x^{5} + 365 x^{4} + 1275 x^{3} - 275 x^{2} + 125 x + 625 \)

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:  \(4229909006359687890625=5^{8}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.47$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{2}{25} a^{9} - \frac{2}{25} a^{8} + \frac{1}{5} a^{7} - \frac{3}{25} a^{6} - \frac{2}{5} a^{5} - \frac{6}{25} a^{4} + \frac{2}{25} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{10} + \frac{2}{25} a^{8} + \frac{7}{25} a^{7} - \frac{7}{25} a^{6} + \frac{9}{25} a^{5} - \frac{7}{25} a^{4} + \frac{3}{25} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{1975} a^{14} + \frac{36}{1975} a^{13} + \frac{37}{1975} a^{12} - \frac{156}{1975} a^{11} + \frac{49}{1975} a^{10} - \frac{7}{1975} a^{9} + \frac{5}{79} a^{8} + \frac{169}{395} a^{7} - \frac{904}{1975} a^{6} - \frac{698}{1975} a^{5} + \frac{254}{1975} a^{4} - \frac{573}{1975} a^{3} + \frac{44}{395} a^{2} - \frac{123}{395} a + \frac{25}{79}$, $\frac{1}{170648520898929875} a^{15} + \frac{37865274506713}{170648520898929875} a^{14} + \frac{1579452625718968}{170648520898929875} a^{13} - \frac{62191955426442}{6825940835957195} a^{12} - \frac{9923907058938381}{170648520898929875} a^{11} - \frac{12868315458248694}{170648520898929875} a^{10} - \frac{3592071569971363}{170648520898929875} a^{9} + \frac{2073529544686304}{170648520898929875} a^{8} + \frac{83663193837147804}{170648520898929875} a^{7} - \frac{4614629141598462}{34129704179785975} a^{6} + \frac{18965436466758889}{170648520898929875} a^{5} + \frac{8501525544530367}{34129704179785975} a^{4} - \frac{15960587701970069}{34129704179785975} a^{3} - \frac{3336703049418908}{6825940835957195} a^{2} + \frac{2396859791485997}{6825940835957195} a + \frac{42947253250986}{1365188167191439}$

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:  \( -1 \) (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:  \( 51535.6216372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 8.8.65037750625.1, 8.0.13007550125.1 x4, 8.0.643938125.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$