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Properties

Label 16.0.10461504735...9344.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{6}\cdot 13^{8}$
Root discriminant $36.62$
Ramified primes $2, 3, 13$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3303, 9876, 16696, -476, -5388, 24, 2596, 892, -831, -264, 312, 52, -68, -12, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 12*x^13 - 68*x^12 + 52*x^11 + 312*x^10 - 264*x^9 - 831*x^8 + 892*x^7 + 2596*x^6 + 24*x^5 - 5388*x^4 - 476*x^3 + 16696*x^2 + 9876*x + 3303)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 12*x^13 - 68*x^12 + 52*x^11 + 312*x^10 - 264*x^9 - 831*x^8 + 892*x^7 + 2596*x^6 + 24*x^5 - 5388*x^4 - 476*x^3 + 16696*x^2 + 9876*x + 3303, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 12 x^{13} - 68 x^{12} + 52 x^{11} + 312 x^{10} - 264 x^{9} - 831 x^{8} + 892 x^{7} + 2596 x^{6} + 24 x^{5} - 5388 x^{4} - 476 x^{3} + 16696 x^{2} + 9876 x + 3303 \)

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:  \(10461504735757671642169344=2^{44}\cdot 3^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.62$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{14} - \frac{1}{9} a^{11} - \frac{2}{27} a^{10} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{27} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{5}{27} a - \frac{4}{9}$, $\frac{1}{12297907209951216875432097} a^{15} - \frac{1795499401532942752420}{455478044813008032423411} a^{14} + \frac{202299333827831343773678}{4099302403317072291810699} a^{13} - \frac{72177327652592233892354}{1366434134439024097270233} a^{12} - \frac{1306403856868019874572957}{12297907209951216875432097} a^{11} + \frac{358138056778645203922006}{4099302403317072291810699} a^{10} + \frac{441821335034335463272087}{4099302403317072291810699} a^{9} + \frac{984165207951622515384172}{4099302403317072291810699} a^{8} + \frac{10451615819631649480943}{49389185582133401106153} a^{7} - \frac{4733508782526067165130669}{12297907209951216875432097} a^{6} - \frac{1377002303555339818718170}{4099302403317072291810699} a^{5} - \frac{63498854707910827375256}{151826014937669344141137} a^{4} + \frac{519189557936632934640970}{4099302403317072291810699} a^{3} - \frac{1802323934681907544992758}{12297907209951216875432097} a^{2} - \frac{666537941489971322693756}{4099302403317072291810699} a + \frac{381134658341337893468077}{1366434134439024097270233}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 41537.5443033 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.4.32448.1, 4.4.8112.1, \(\Q(\sqrt{2}, \sqrt{13})\), 8.0.808606236672.9 x2, 8.0.202151559168.8 x2, 8.8.16845963264.1

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
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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$