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Properties

Label 16.0.43190748110...8641.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 37^{12}$
Root discriminant $25.98$
Ramified primes $3, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, -47, 659, 14, -227, 18, 878, -55, 148, 13, -100, 18, 55, -8, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 7*x^14 - 8*x^13 + 55*x^12 + 18*x^11 - 100*x^10 + 13*x^9 + 148*x^8 - 55*x^7 + 878*x^6 + 18*x^5 - 227*x^4 + 14*x^3 + 659*x^2 - 47*x + 2209)
 
gp: K = bnfinit(x^16 - x^15 - 7*x^14 - 8*x^13 + 55*x^12 + 18*x^11 - 100*x^10 + 13*x^9 + 148*x^8 - 55*x^7 + 878*x^6 + 18*x^5 - 227*x^4 + 14*x^3 + 659*x^2 - 47*x + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 7 x^{14} - 8 x^{13} + 55 x^{12} + 18 x^{11} - 100 x^{10} + 13 x^{9} + 148 x^{8} - 55 x^{7} + 878 x^{6} + 18 x^{5} - 227 x^{4} + 14 x^{3} + 659 x^{2} - 47 x + 2209 \)

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:  \(43190748110316471478641=3^{8}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.98$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{2772} a^{12} - \frac{4}{99} a^{11} - \frac{1}{84} a^{10} - \frac{13}{231} a^{9} - \frac{1}{42} a^{8} + \frac{1279}{2772} a^{7} - \frac{293}{1386} a^{6} - \frac{67}{396} a^{5} + \frac{37}{77} a^{4} - \frac{104}{231} a^{3} - \frac{145}{308} a^{2} + \frac{199}{693} a - \frac{41}{2772}$, $\frac{1}{2772} a^{13} - \frac{103}{2772} a^{11} + \frac{17}{154} a^{10} + \frac{1}{154} a^{9} + \frac{355}{2772} a^{8} + \frac{61}{462} a^{7} - \frac{5}{396} a^{6} + \frac{505}{1386} a^{5} - \frac{23}{77} a^{4} + \frac{97}{924} a^{3} - \frac{305}{693} a^{2} - \frac{89}{252} a - \frac{97}{198}$, $\frac{1}{5544} a^{14} - \frac{1}{5544} a^{13} + \frac{47}{616} a^{11} + \frac{157}{1848} a^{10} - \frac{23}{5544} a^{9} - \frac{29}{504} a^{8} - \frac{215}{693} a^{7} + \frac{95}{1848} a^{6} + \frac{229}{616} a^{5} + \frac{829}{1848} a^{4} - \frac{2543}{5544} a^{3} - \frac{97}{2772} a^{2} - \frac{11}{56} a + \frac{431}{1848}$, $\frac{1}{198423293379696} a^{15} + \frac{945088729}{33070548896616} a^{14} - \frac{366998489}{6012827072112} a^{13} - \frac{5900081161}{66141097793232} a^{12} - \frac{1677178163327}{99211646689848} a^{11} - \frac{3181670618681}{49605823344924} a^{10} - \frac{486898131519}{5511758149436} a^{9} + \frac{29890651617389}{198423293379696} a^{8} - \frac{608562921667}{1562388136848} a^{7} - \frac{23863263547789}{99211646689848} a^{6} + \frac{1023879899845}{4509620304084} a^{5} + \frac{6745706612585}{99211646689848} a^{4} - \frac{74409977494009}{198423293379696} a^{3} + \frac{8554976399669}{28346184768528} a^{2} + \frac{6952186424377}{24802911672462} a - \frac{1826156856421}{4221772199568}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{84994734}{125267230669} a^{15} + \frac{4364917079}{1503206768028} a^{14} + \frac{5266079047}{1503206768028} a^{13} - \frac{3340783099}{250534461338} a^{12} - \frac{13089381145}{214743824004} a^{11} + \frac{184944261269}{1503206768028} a^{10} + \frac{278833468055}{1503206768028} a^{9} - \frac{548120220259}{1503206768028} a^{8} - \frac{1860715111}{5918136882} a^{7} + \frac{1285836750883}{1503206768028} a^{6} - \frac{46465392751}{214743824004} a^{5} + \frac{1008835412765}{1503206768028} a^{4} + \frac{724538043905}{1503206768028} a^{3} - \frac{87694528379}{125267230669} a^{2} - \frac{516150453069}{501068922676} a + \frac{108843520025}{31983122724} \) (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:  \( 96123.0350705 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{37})\), 4.2.4107.1 x2, 4.0.333.1 x2, 4.0.455877.1 x2, 4.2.151959.1 x2, 4.4.455877.1, 4.0.50653.1, 8.0.151807041.1, 8.0.207823839129.2, 8.0.207823839129.1, 8.4.207823839129.1 x2, 8.0.23091537681.1 x2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$