↑

Properties

Label 16.0.13852994719059873.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 13^{4}\cdot 97^{3}$
Root discriminant $10.21$
Ramified primes $3, 13, 97$
Class number $1$
Class group Trivial
Galois group 16T1432

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 20, 45, 77, 105, 113, 87, 49, 18, -5, -9, -7, -3, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + x^14 - 3*x^13 - 7*x^12 - 9*x^11 - 5*x^10 + 18*x^9 + 49*x^8 + 87*x^7 + 113*x^6 + 105*x^5 + 77*x^4 + 45*x^3 + 20*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 + x^14 - 3*x^13 - 7*x^12 - 9*x^11 - 5*x^10 + 18*x^9 + 49*x^8 + 87*x^7 + 113*x^6 + 105*x^5 + 77*x^4 + 45*x^3 + 20*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + x^{14} - 3 x^{13} - 7 x^{12} - 9 x^{11} - 5 x^{10} + 18 x^{9} + 49 x^{8} + 87 x^{7} + 113 x^{6} + 105 x^{5} + 77 x^{4} + 45 x^{3} + 20 x^{2} + 6 x + 1 \)

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:  \(13852994719059873=3^{12}\cdot 13^{4}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.21$
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:  $3, 13, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10993277} a^{15} + \frac{673840}{10993277} a^{14} + \frac{5025670}{10993277} a^{13} - \frac{3493607}{10993277} a^{12} + \frac{1175724}{10993277} a^{11} - \frac{2633408}{10993277} a^{10} - \frac{4846493}{10993277} a^{9} + \frac{962011}{10993277} a^{8} + \frac{927430}{10993277} a^{7} + \frac{4613668}{10993277} a^{6} - \frac{2703813}{10993277} a^{5} + \frac{431949}{10993277} a^{4} - \frac{4480892}{10993277} a^{3} - \frac{1797692}{10993277} a^{2} + \frac{3408647}{10993277} a + \frac{2364491}{10993277}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{37191}{32621} a^{15} - \frac{6221}{32621} a^{14} + \frac{40156}{32621} a^{13} - \frac{120581}{32621} a^{12} - \frac{239819}{32621} a^{11} - \frac{298345}{32621} a^{10} - \frac{151471}{32621} a^{9} + \frac{710520}{32621} a^{8} + \frac{1702725}{32621} a^{7} + \frac{3011026}{32621} a^{6} + \frac{3736732}{32621} a^{5} + \frac{3307078}{32621} a^{4} + \frac{2309296}{32621} a^{3} + \frac{1190282}{32621} a^{2} + \frac{467354}{32621} a + \frac{115483}{32621} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53.4955230971 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 119 conjugacy class representatives for t16n1432 are not computed
Character table for t16n1432 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 8.0.1327833.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
3Data not computed
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed