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Properties

Label 16.0.13872985106...121.12
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 47^{8}$
Root discriminant $57.40$
Ramified primes $17, 47$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2032643, -6329080, 9697143, -9559647, 6687334, -3644405, 1657820, -657477, 237086, -66039, 17733, -3491, 852, -114, 28, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 28*x^14 - 114*x^13 + 852*x^12 - 3491*x^11 + 17733*x^10 - 66039*x^9 + 237086*x^8 - 657477*x^7 + 1657820*x^6 - 3644405*x^5 + 6687334*x^4 - 9559647*x^3 + 9697143*x^2 - 6329080*x + 2032643)
 
gp: K = bnfinit(x^16 - 3*x^15 + 28*x^14 - 114*x^13 + 852*x^12 - 3491*x^11 + 17733*x^10 - 66039*x^9 + 237086*x^8 - 657477*x^7 + 1657820*x^6 - 3644405*x^5 + 6687334*x^4 - 9559647*x^3 + 9697143*x^2 - 6329080*x + 2032643, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 28 x^{14} - 114 x^{13} + 852 x^{12} - 3491 x^{11} + 17733 x^{10} - 66039 x^{9} + 237086 x^{8} - 657477 x^{7} + 1657820 x^{6} - 3644405 x^{5} + 6687334 x^{4} - 9559647 x^{3} + 9697143 x^{2} - 6329080 x + 2032643 \)

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:  \(13872985106194361214049869121=17^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.40$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{312} a^{14} + \frac{5}{24} a^{13} + \frac{11}{104} a^{12} + \frac{11}{312} a^{11} + \frac{49}{312} a^{10} - \frac{1}{6} a^{9} + \frac{19}{156} a^{8} + \frac{47}{104} a^{7} - \frac{4}{39} a^{6} + \frac{19}{39} a^{5} + \frac{37}{78} a^{4} - \frac{133}{312} a^{3} + \frac{25}{52} a^{2} - \frac{1}{6} a - \frac{107}{312}$, $\frac{1}{2691724496223184243473897768852827425952162736} a^{15} + \frac{263093734248923726568071428011211742586857}{224310374685265353622824814071068952162680228} a^{14} + \frac{11487967915587054904845824425007405369915681}{672931124055796060868474442213206856488040684} a^{13} - \frac{7988189911766693222614506037614219257972609}{1345862248111592121736948884426413712976081368} a^{12} + \frac{27185301213680542534776340587208458039370955}{448620749370530707245649628142137904325360456} a^{11} - \frac{38266795620772673365713744254176549122038403}{897241498741061414491299256284275808650720912} a^{10} - \frac{297030435398311949550733992061120953648675895}{1345862248111592121736948884426413712976081368} a^{9} - \frac{816081800977360370545434397757429471979667489}{2691724496223184243473897768852827425952162736} a^{8} + \frac{1089013123575901095745372600514093323978475647}{2691724496223184243473897768852827425952162736} a^{7} - \frac{43377971236462833531351334908457037581106979}{112155187342632676811412407035534476081340114} a^{6} - \frac{18465321868139115737759789656444847566014807}{224310374685265353622824814071068952162680228} a^{5} + \frac{10071326117217352017465773703203859579166573}{897241498741061414491299256284275808650720912} a^{4} + \frac{946644022523531902372826805200456259135120023}{2691724496223184243473897768852827425952162736} a^{3} + \frac{590855886648904219168905350805684586999718731}{1345862248111592121736948884426413712976081368} a^{2} - \frac{6395056943173128314579856332364710552697593}{16929084881906819141345268986495769974541904} a + \frac{87089816423308673901289951666462200744550471}{2691724496223184243473897768852827425952162736}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( 439805.205951 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

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 $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.13583.1 x2, 4.0.37553.1 x2, 8.0.407555836801.2, 8.2.2506034826287.2 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$