↑

Properties

Label 16.4.40745431714...4569.9
Degree $16$
Signature $[4, 6]$
Discriminant $13^{12}\cdot 53^{10}$
Root discriminant $81.87$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-239067, -998151, -1147948, -242564, -19090, 23057, 4204, 10716, 3107, -1315, -290, 160, 97, -46, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 - 46*x^13 + 97*x^12 + 160*x^11 - 290*x^10 - 1315*x^9 + 3107*x^8 + 10716*x^7 + 4204*x^6 + 23057*x^5 - 19090*x^4 - 242564*x^3 - 1147948*x^2 - 998151*x - 239067)
 
gp: K = bnfinit(x^16 - 2*x^15 - 16*x^14 - 46*x^13 + 97*x^12 + 160*x^11 - 290*x^10 - 1315*x^9 + 3107*x^8 + 10716*x^7 + 4204*x^6 + 23057*x^5 - 19090*x^4 - 242564*x^3 - 1147948*x^2 - 998151*x - 239067, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 16 x^{14} - 46 x^{13} + 97 x^{12} + 160 x^{11} - 290 x^{10} - 1315 x^{9} + 3107 x^{8} + 10716 x^{7} + 4204 x^{6} + 23057 x^{5} - 19090 x^{4} - 242564 x^{3} - 1147948 x^{2} - 998151 x - 239067 \)

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:  \(4074543171431096342049568754569=13^{12}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.87$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{4}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{28888644206523933954647742088054860451137} a^{15} - \frac{914832363209283876485751942170593821052}{28888644206523933954647742088054860451137} a^{14} - \frac{3740321360998881231660953104811291370665}{28888644206523933954647742088054860451137} a^{13} + \frac{1151751755083695948021302618521868492777}{3209849356280437106071971343117206716793} a^{12} + \frac{9849027920016644968587781496112977599582}{28888644206523933954647742088054860451137} a^{11} - \frac{9974173735873290307497042579857513184685}{28888644206523933954647742088054860451137} a^{10} + \frac{308881966827453231141961935853504547866}{9629548068841311318215914029351620150379} a^{9} - \frac{30265082792761765985641667261255763319}{114184364452663770571730205881639764629} a^{8} + \frac{11738129172009421381283113029136763705128}{28888644206523933954647742088054860451137} a^{7} - \frac{3519063146693358206011022397650690049089}{28888644206523933954647742088054860451137} a^{6} - \frac{968884389297295262966835188519337692998}{28888644206523933954647742088054860451137} a^{5} - \frac{9124370832724424792577867283310199636134}{28888644206523933954647742088054860451137} a^{4} + \frac{1405631939524068533215969304500118786224}{3209849356280437106071971343117206716793} a^{3} + \frac{4911375022937618295538690299139238872339}{28888644206523933954647742088054860451137} a^{2} - \frac{1334825050948256280216062946389500784656}{9629548068841311318215914029351620150379} a + \frac{316013573757854950826167245521532596236}{3209849356280437106071971343117206716793}$

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:  $9$
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:  \( 1119695463.09 \) (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, 8.4.38085844705129.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed