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Properties

Label 16.0.13066428858...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $20.88$
Ramified primes $3, 5, 13$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -192, -528, -916, 3049, -534, -2341, 717, 768, -95, -200, -15, 32, 1, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 + x^13 + 32*x^12 - 15*x^11 - 200*x^10 - 95*x^9 + 768*x^8 + 717*x^7 - 2341*x^6 - 534*x^5 + 3049*x^4 - 916*x^3 - 528*x^2 - 192*x + 256)
 
gp: K = bnfinit(x^16 - x^15 - x^14 + x^13 + 32*x^12 - 15*x^11 - 200*x^10 - 95*x^9 + 768*x^8 + 717*x^7 - 2341*x^6 - 534*x^5 + 3049*x^4 - 916*x^3 - 528*x^2 - 192*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - x^{14} + x^{13} + 32 x^{12} - 15 x^{11} - 200 x^{10} - 95 x^{9} + 768 x^{8} + 717 x^{7} - 2341 x^{6} - 534 x^{5} + 3049 x^{4} - 916 x^{3} - 528 x^{2} - 192 x + 256 \)

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:  \(1306642885859619140625=3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.88$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{2} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{5}{16} a^{4} - \frac{3}{8} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{4009096735057030976} a^{15} + \frac{56322118065428243}{4009096735057030976} a^{14} - \frac{7067250735702487}{211005091318791104} a^{13} - \frac{123507721362546739}{4009096735057030976} a^{12} - \frac{141140135049035899}{1002274183764257744} a^{11} + \frac{343306701836393681}{4009096735057030976} a^{10} + \frac{224818431925227307}{1002274183764257744} a^{9} + \frac{1093883822737110945}{4009096735057030976} a^{8} - \frac{23510158227531627}{1002274183764257744} a^{7} - \frac{563421825565011411}{4009096735057030976} a^{6} + \frac{1570901574208972159}{4009096735057030976} a^{5} + \frac{8306127130094945}{105502545659395552} a^{4} + \frac{1397703111280492081}{4009096735057030976} a^{3} + \frac{29112033408919693}{62642136485266109} a^{2} + \frac{21626258515897709}{125284272970532218} a - \frac{14588115386573971}{62642136485266109}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{1216647377039}{5626652391104} a^{15} - \frac{227360781295}{5626652391104} a^{14} - \frac{1440513153855}{5626652391104} a^{13} + \frac{1530931135}{5626652391104} a^{12} + \frac{2432077505025}{351665774444} a^{11} + \frac{13398251695695}{5626652391104} a^{10} - \frac{29211983259075}{703331548888} a^{9} - \frac{307645730737185}{5626652391104} a^{8} + \frac{21522114990105}{175832887222} a^{7} + \frac{1449079612278195}{5626652391104} a^{6} - \frac{1668870672261771}{5626652391104} a^{5} - \frac{1026621395943805}{2813326195552} a^{4} + \frac{2023076095207095}{5626652391104} a^{3} + \frac{144926153086885}{1406663097776} a^{2} - \frac{2336461197510}{87916443611} a - \frac{5718543570829}{87916443611} \) (order $6$)
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:  \( 5639.28109272 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.4.190125.1, 4.0.21125.1, 4.0.2925.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.117.1, 8.0.8555625.1, 8.0.213890625.1, 8.0.36147515625.2

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$