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Properties

Label 16.0.11059281013...5321.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{12}\cdot 19^{14}$
Root discriminant $56.59$
Ramified primes $7, 19$
Class number $1$ (GRH)
Class group Trivial (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![11519, -2819, 32352, 52783, 54986, 2780, -17082, 6707, 7725, 1862, 792, -142, 19, 67, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 67 x^{13} + 19 x^{12} - 142 x^{11} + 792 x^{10} + 1862 x^{9} + 7725 x^{8} + 6707 x^{7} - 17082 x^{6} + 2780 x^{5} + 54986 x^{4} + 52783 x^{3} + 32352 x^{2} - 2819 x + 11519 \)

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:  \(11059281013440062648863435321=7^{12}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.59$
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:  $7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{63} a^{12} + \frac{1}{9} a^{11} - \frac{1}{7} a^{10} + \frac{4}{63} a^{9} + \frac{5}{63} a^{8} - \frac{3}{7} a^{7} + \frac{10}{21} a^{6} + \frac{1}{21} a^{5} + \frac{5}{63} a^{4} - \frac{1}{9} a^{3} - \frac{10}{21} a^{2} - \frac{16}{63} a - \frac{20}{63}$, $\frac{1}{2457} a^{13} + \frac{1}{819} a^{12} + \frac{5}{2457} a^{11} + \frac{124}{2457} a^{10} - \frac{32}{2457} a^{9} + \frac{79}{2457} a^{8} - \frac{353}{819} a^{7} - \frac{20}{273} a^{6} + \frac{116}{351} a^{5} - \frac{73}{273} a^{4} + \frac{523}{2457} a^{3} + \frac{335}{2457} a^{2} - \frac{880}{2457} a + \frac{899}{2457}$, $\frac{1}{100737} a^{14} - \frac{5}{100737} a^{13} - \frac{643}{100737} a^{12} + \frac{49}{1599} a^{11} + \frac{16}{351} a^{10} + \frac{5210}{100737} a^{9} + \frac{14026}{100737} a^{8} + \frac{1828}{33579} a^{7} + \frac{40043}{100737} a^{6} - \frac{40966}{100737} a^{5} - \frac{21911}{100737} a^{4} - \frac{13750}{33579} a^{3} - \frac{9410}{100737} a^{2} + \frac{42493}{100737} a - \frac{8635}{100737}$, $\frac{1}{63564723414784127062751226093} a^{15} + \frac{59976605568264082931483}{21188241138261375687583742031} a^{14} - \frac{80862600839967491089855}{492749793913055248548459117} a^{13} + \frac{299965039994055332868143758}{63564723414784127062751226093} a^{12} - \frac{1129170468815621166914153936}{21188241138261375687583742031} a^{11} + \frac{10500063881467728683662817723}{63564723414784127062751226093} a^{10} + \frac{6043254496681019637764770552}{63564723414784127062751226093} a^{9} + \frac{6405592473091528713021705829}{63564723414784127062751226093} a^{8} + \frac{11249190254627149849999300364}{63564723414784127062751226093} a^{7} + \frac{8857635305328175265323303847}{21188241138261375687583742031} a^{6} + \frac{4448077493365414613008146169}{21188241138261375687583742031} a^{5} + \frac{18859591748160269839971702797}{63564723414784127062751226093} a^{4} - \frac{441193628334803554088300761}{1008963863726732175599225811} a^{3} - \frac{25428110143087945145021010923}{63564723414784127062751226093} a^{2} - \frac{31655369450725224470940484639}{63564723414784127062751226093} a - \frac{16191678071563129322326480807}{63564723414784127062751226093}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 25363590.094 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-7}, \sqrt{-19})\), 4.2.336091.1 x2, 4.0.48013.1 x2, 8.0.112957160281.1, 8.2.105163116221611.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
19Data not computed