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Properties

Label 16.0.40073498016...161.16
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{12}$
Root discriminant $70.82$
Ramified primes $13, 53$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37181, 5183, -19619, 39331, 1080, -23827, 9176, 3502, -2986, 123, 752, -568, 248, -77, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 24*x^14 - 77*x^13 + 248*x^12 - 568*x^11 + 752*x^10 + 123*x^9 - 2986*x^8 + 3502*x^7 + 9176*x^6 - 23827*x^5 + 1080*x^4 + 39331*x^3 - 19619*x^2 + 5183*x + 37181)
 
gp: K = bnfinit(x^16 - 7*x^15 + 24*x^14 - 77*x^13 + 248*x^12 - 568*x^11 + 752*x^10 + 123*x^9 - 2986*x^8 + 3502*x^7 + 9176*x^6 - 23827*x^5 + 1080*x^4 + 39331*x^3 - 19619*x^2 + 5183*x + 37181, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 24 x^{14} - 77 x^{13} + 248 x^{12} - 568 x^{11} + 752 x^{10} + 123 x^{9} - 2986 x^{8} + 3502 x^{7} + 9176 x^{6} - 23827 x^{5} + 1080 x^{4} + 39331 x^{3} - 19619 x^{2} + 5183 x + 37181 \)

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:  \(400734980167009195224860426161=13^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.82$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{4}{13} a^{10} + \frac{1}{13} a^{9} + \frac{3}{13} a^{8} - \frac{4}{13} a^{7} - \frac{1}{13} a^{6} - \frac{5}{13} a^{5} - \frac{2}{13} a^{2} - \frac{6}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{12} - \frac{2}{13} a^{10} - \frac{6}{13} a^{9} - \frac{5}{13} a^{8} - \frac{4}{13} a^{7} + \frac{4}{13} a^{6} + \frac{6}{13} a^{5} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} + \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{13} - \frac{1}{13} a^{10} - \frac{3}{13} a^{9} + \frac{2}{13} a^{8} - \frac{4}{13} a^{7} + \frac{4}{13} a^{6} + \frac{3}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} + \frac{2}{13} a^{2} + \frac{4}{13} a - \frac{5}{13}$, $\frac{1}{169} a^{14} - \frac{5}{169} a^{13} + \frac{4}{169} a^{12} - \frac{6}{169} a^{11} - \frac{12}{169} a^{10} - \frac{64}{169} a^{9} - \frac{49}{169} a^{8} - \frac{24}{169} a^{7} - \frac{74}{169} a^{6} - \frac{33}{169} a^{5} - \frac{17}{169} a^{4} - \frac{40}{169} a^{3} - \frac{3}{13} a^{2} + \frac{3}{169} a + \frac{17}{169}$, $\frac{1}{37315548897559812712541244523} a^{15} + \frac{71461358150815777856353517}{37315548897559812712541244523} a^{14} + \frac{745005912084538362830470710}{37315548897559812712541244523} a^{13} + \frac{964371895081290175597641714}{37315548897559812712541244523} a^{12} - \frac{677967036101984577444812401}{37315548897559812712541244523} a^{11} - \frac{3612032150049628693180281517}{37315548897559812712541244523} a^{10} - \frac{710928653945227505079156834}{2870426838273831747118557271} a^{9} - \frac{12133023114050540283806849437}{37315548897559812712541244523} a^{8} + \frac{18242450804011097311131370690}{37315548897559812712541244523} a^{7} - \frac{2778623524214327832569475188}{37315548897559812712541244523} a^{6} + \frac{4066009328830552051298462493}{37315548897559812712541244523} a^{5} + \frac{12278463456434639728848060735}{37315548897559812712541244523} a^{4} + \frac{14418785895088984971715819006}{37315548897559812712541244523} a^{3} - \frac{15321535013427732140756151444}{37315548897559812712541244523} a^{2} + \frac{15463526323052139305211763533}{37315548897559812712541244523} a - \frac{11393282453820930081578339114}{37315548897559812712541244523}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 16858689.275 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.48695101400413.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$