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Properties

Label 16.4.24109722907...601.12
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-39, 104, -127, 1136, -2139, -1178, 4908, -2436, -1703, 2216, -870, -100, 320, -168, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 320*x^12 - 100*x^11 - 870*x^10 + 2216*x^9 - 1703*x^8 - 2436*x^7 + 4908*x^6 - 1178*x^5 - 2139*x^4 + 1136*x^3 - 127*x^2 + 104*x - 39)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 320*x^12 - 100*x^11 - 870*x^10 + 2216*x^9 - 1703*x^8 - 2436*x^7 + 4908*x^6 - 1178*x^5 - 2139*x^4 + 1136*x^3 - 127*x^2 + 104*x - 39, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 320 x^{12} - 100 x^{11} - 870 x^{10} + 2216 x^{9} - 1703 x^{8} - 2436 x^{7} + 4908 x^{6} - 1178 x^{5} - 2139 x^{4} + 1136 x^{3} - 127 x^{2} + 104 x - 39 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24109722907876309716269637601=13^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.41$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{26} a^{12} - \frac{3}{13} a^{11} - \frac{1}{13} a^{10} - \frac{1}{26} a^{8} - \frac{5}{13} a^{7} + \frac{7}{26} a^{6} - \frac{11}{26} a^{5} - \frac{9}{26} a^{4} - \frac{2}{13} a^{3} - \frac{2}{13} a^{2}$, $\frac{1}{26} a^{13} + \frac{1}{26} a^{11} + \frac{1}{26} a^{10} - \frac{1}{26} a^{9} - \frac{3}{26} a^{8} + \frac{6}{13} a^{7} - \frac{4}{13} a^{6} - \frac{5}{13} a^{5} + \frac{7}{26} a^{4} - \frac{1}{13} a^{3} + \frac{1}{13} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{913507894} a^{14} - \frac{7}{913507894} a^{13} - \frac{10905813}{913507894} a^{12} + \frac{65434969}{913507894} a^{11} + \frac{4572028}{456753947} a^{10} - \frac{188787049}{913507894} a^{9} - \frac{170238467}{913507894} a^{8} - \frac{207062575}{913507894} a^{7} - \frac{167939862}{456753947} a^{6} - \frac{328437117}{913507894} a^{5} - \frac{7957387}{35134919} a^{4} - \frac{184971447}{456753947} a^{3} + \frac{3496570}{35134919} a^{2} + \frac{21722617}{70269838} a - \frac{18438235}{70269838}$, $\frac{1}{913507894} a^{15} - \frac{5452931}{456753947} a^{13} - \frac{5452861}{456753947} a^{12} + \frac{401342}{35134919} a^{11} - \frac{124778657}{913507894} a^{10} - \frac{121485969}{913507894} a^{9} - \frac{28470003}{913507894} a^{8} - \frac{207527954}{456753947} a^{7} + \frac{60928497}{913507894} a^{6} + \frac{234571801}{913507894} a^{5} + \frac{4414230}{456753947} a^{4} + \frac{120917122}{456753947} a^{3} + \frac{404759}{70269838} a^{2} - \frac{3459796}{35134919} a - \frac{11831444}{35134919}$

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:  $9$
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:  \( 285427569.891 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
53Data not computed