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Properties

Label 16.0.44016602739...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 101^{12}$
Root discriminant $71.24$
Ramified primes $5, 101$
Class number $100$ (GRH)
Class group $[10, 10]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1849, -21672, 138810, -79769, -85833, 88051, 7604, -38510, 12140, 4185, -2514, -93, 180, 9, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 12*x^14 + 9*x^13 + 180*x^12 - 93*x^11 - 2514*x^10 + 4185*x^9 + 12140*x^8 - 38510*x^7 + 7604*x^6 + 88051*x^5 - 85833*x^4 - 79769*x^3 + 138810*x^2 - 21672*x + 1849)
 
gp: K = bnfinit(x^16 - x^15 - 12*x^14 + 9*x^13 + 180*x^12 - 93*x^11 - 2514*x^10 + 4185*x^9 + 12140*x^8 - 38510*x^7 + 7604*x^6 + 88051*x^5 - 85833*x^4 - 79769*x^3 + 138810*x^2 - 21672*x + 1849, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 12 x^{14} + 9 x^{13} + 180 x^{12} - 93 x^{11} - 2514 x^{10} + 4185 x^{9} + 12140 x^{8} - 38510 x^{7} + 7604 x^{6} + 88051 x^{5} - 85833 x^{4} - 79769 x^{3} + 138810 x^{2} - 21672 x + 1849 \)

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:  \(440166027395300672133281640625=5^{8}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.24$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{505} a^{12} + \frac{209}{505} a^{11} + \frac{4}{101} a^{10} - \frac{68}{505} a^{9} - \frac{131}{505} a^{8} - \frac{193}{505} a^{7} + \frac{24}{505} a^{6} + \frac{47}{505} a^{5} + \frac{184}{505} a^{4} + \frac{232}{505} a^{3} + \frac{3}{505} a^{2} - \frac{227}{505} a + \frac{84}{505}$, $\frac{1}{505} a^{13} - \frac{231}{505} a^{11} - \frac{208}{505} a^{10} - \frac{59}{505} a^{9} - \frac{84}{505} a^{8} - \frac{39}{505} a^{7} + \frac{81}{505} a^{6} - \frac{44}{505} a^{5} + \frac{156}{505} a^{4} - \frac{1}{101} a^{3} + \frac{156}{505} a^{2} + \frac{57}{505} a + \frac{119}{505}$, $\frac{1}{6565} a^{14} + \frac{1}{6565} a^{13} - \frac{1}{1313} a^{12} + \frac{370}{1313} a^{11} - \frac{292}{6565} a^{10} + \frac{2164}{6565} a^{9} - \frac{439}{6565} a^{8} - \frac{3176}{6565} a^{7} - \frac{2114}{6565} a^{6} + \frac{129}{6565} a^{5} + \frac{267}{1313} a^{4} - \frac{947}{6565} a^{3} + \frac{1901}{6565} a^{2} - \frac{1636}{6565} a + \frac{418}{6565}$, $\frac{1}{4676228004014482658739499945} a^{15} + \frac{234991281433461818071163}{4676228004014482658739499945} a^{14} - \frac{87269370418912705706}{106077807862769835508915} a^{13} - \frac{4328923541248813251637147}{4676228004014482658739499945} a^{12} + \frac{16360316820489530700096059}{935245600802896531747899989} a^{11} + \frac{155133383405299903423040393}{935245600802896531747899989} a^{10} - \frac{450783918383459326150599539}{935245600802896531747899989} a^{9} + \frac{1920438405528363152988632893}{4676228004014482658739499945} a^{8} + \frac{264979389051428029647491950}{935245600802896531747899989} a^{7} - \frac{1973740611972354951950234532}{4676228004014482658739499945} a^{6} - \frac{907850145579583483328443726}{4676228004014482658739499945} a^{5} - \frac{404126861028064296963303698}{935245600802896531747899989} a^{4} + \frac{26063292892572713665326636}{359709846462652512210730765} a^{3} + \frac{302450974331257982554716437}{935245600802896531747899989} a^{2} + \frac{292544329071213214863997589}{935245600802896531747899989} a + \frac{44936563222424811629156201}{108749488465453085086965115}$

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

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  \( 2201007.40426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 4.0.1030301.1, 4.0.5151505.1, 8.4.6568812813125.3, 8.4.6568812813125.1, 8.0.26538003765025.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$